How to distinguish between the spectrum of an atom in motion and the one of a scaled atom? Galaxies are moving dragged by the space expansion. When atoms are in motion the doppler effect will shift the spectra of the emitted photons.   
The proton-to-electron mass ratio, $\frac{m_e}{m_p}$ has been measured constant along the history of the universe, but nothing was said about the constancy of the electron or proton's masses.  
The photon's energy obey the Sommerfeld relation, $E_{jn}=-m_e*f(j,n,\alpha,c)$, as seen here, and it is evident that a shifted (1) spectrum is obtained with a larger $m_e$.
The spectra lines are not only due to the Hydrogen atom; there are other spectral lines due to molecular interactions, due to electric/magnetic dipoles, etc, and so the electromagnetic interaction,the Coulomb's law, $F_{}=\frac{1}{4\cdot \pi\cdot \varepsilon}\cdot \frac{q1\cdot q2}{d^2}$ must be analyzed.  
If we scale all masses by the relation $\alpha(t)$ (not related with the above fine structure constant), where $t$ is time (past), and also scale the charges and the distances and time by the same factor, gives exactly the same value $F_{}=\frac{1}{4\cdot \pi\cdot \varepsilon}\cdot \frac{q_1\cdot q_2\cdot \alpha^ 2(t)}{d^2\cdot \alpha^2(t)}$. Thus the system with and without the transformation behaves in the same manner. The same procedure shows that the universal gravitational law is also insensitive to the scaling of the atom (2). This should not be a complete surprise because the scaling of masses, charges, time units and distances is routinely used on computer simulations that mimic the universe in a consistent way.   
The conclusion is that there is no easy way to distinguish between the spectrum of an atom in motion and the one of a scaled atom. 
The photons that were emitted by a larger atom in the past are received now without any change on its wavelength (here I'm assuming a static space). 
The mainstream viewpoint, not being aware that scaling the atom gave the same observational results, adopted the receding interpretation long time ago. As a consequence the models derived from that interpretation (BB, Inflation, DE, DM, ) do not obey the general laws of the universe, namely the energy conservation principle.  
My viewpoint offers a cause for the space expansion.  Most physicists are comfortable with: 'space expands', period, without a known cause.    
Physics is about causes and whys, backed by proper references. 
I used the most basic laws of mainstream physics to show that another viewpoint is inscribed in the laws of nature. 
The question is strict, formulated in the title: How can someone make a distinction between those spectra?  
I've already opted for an answer, presented below to provide context, but that is not for discussion. 
When I graduated as electronic engineer, long time ago, I accepted naively that the fields (electrostatic and gravitational) are sourced by the particles, and expand at $c$ speed, without being drained. But now, older but not senile, I assume without exception, that in the universe there are no 'free lunches' and thus the energy must be transferred from the particles (shrinking) to the fields (growing).  
This new viewpoint is formalized and compared to the $\Lambda CDM$ model in a rigourous document, with the derivation of the scale relation $\alpha(t)$ that corresponds to the universe's evolution, at:
A self-similar model of the Universe unveils the nature of dark energy
preceded by older documents at arxiv:
Cosmological Principle and Relativity - Part I
A relativistic time variation of matter/space fits both local and cosmic data 
Again, my question is:
Can someone provide a way to distinguish between the spectrum of an atom in motion and the one of a scaled atom ? 
maybe by probing the atom's nucleus and find the isotope ratio's abundance, D/H evolution and other isotopes as Mr Webb did with Mg (1998 paper) when in search of the $\alpha$ variability.    
PS: To simplify the argument this question skips some details and to avoid misrepresentations a short resumé of my position is annexed. 
(1) - after the details are done it is a redshifted spectrum at the reception 
(2) - LMTQ units scale equally by $\alpha(t)$, $c,G,\varepsilon$ and the fine structure constant $\alpha$ are invariant. 

A short introduction to the self-similar dilation model
The self-similar model arises from an analysis of a fundamental question: is it the space that expands or standard length unit that decreases? That analysis is not an alternative cosmological model derived from some new hypothesis; on the contrary, it does not depend on hypotheses, it has no parameters besides Hubble parameter; it is simply the identification of the phenomenon behind the data, obtained by deduction from consensual observational results. The phenomenon identified is the following: in invariant space, matter is transforming in field in a self-similar way, feeding field expansion. As a consequence of this phenomenon, matter and field evanesce while field expands since the moment when matter appeared. As we use units in which matter is invariant, i.e., units intrinsic to matter, we cannot detect locally the evanescence of matter; but, as a consequence of our decreasing units, we detect an expanding space. So, like the explanation for the apparent rotation of the cosmic bodies, also the explanation for another global cosmic phenomenon (the apparent receding of the cosmic bodies) lays in us. 
In units where space is invariant, named Space or S units, matter and field evanesce: bodies decrease in size, the velocity of atomic phenomena increases because light speed is invariant but distances within bounded systems of particles decrease. In standard units, intrinsic to matter, here called Atomic or A units, matter and all its phenomena have invariant properties; however, the distance between non-bounded bodies increases, and the wavelength of distant radiations is red-shifted (they were emitted when atoms were greater). The ratios between Atomic and Space units, represented by M for mass, Q for charge, L for length and T for Time, are the following:
$$M=L=T=Q=\alpha(t)$$
The scaling function $\alpha(t)$ is exponential in S units, as is typical in self-similar phenomena:    $$\alpha(t_S)=e^{-H_0 \cdot t_S}$$  
Mass and charge decrease exponentially in S units, and the size of atoms decrease at the same ratio, implying that the phenomena runs faster in the inverse ratio; as A units are such that hold invariant the measures of mass, charge, length of bodies (Einstein concept of reference body) and light speed, they vary all with the same ratio in relation to S units. In A units, space appears to expand at the inverse ratio of the decrease of A length unit; the space scale factor in A units, a, is: $$a=1+H_0 \cdot t_A$$
Therefore, space expands linearly in A units.
In what concerns physical laws, those that do not depend on time or space (local laws), like Planck law, are not affected by the evolution of matter/field and hold the same in both systems of units. The laws for static field relate field and its source in the same moment; as both vary at the same ratio, their relation holds invariant and so the laws – the classic laws are valid both in A and S units. The electromagnetic induction laws can be treated as if they were local, therefore holding valid in both systems, and then consider that, in S, the energy of waves decreases with the square of field evanescence (the energy of electromagnetic waves is proportional to the square of the field). In A, due to the relationship between units, the energy decreases at the inverse ratio of space expansion while the wavelength increases proportionally. This decrease of the energy of the waves (or of the photons) is a mystery for an A observer because in A it is supposed to exist energy conservation, which is violated by electromagnetic waves. The phenomenon is observed in the cosmic microwave background (CMB) because the temperature shift of a Planck radiation implies a decrease of the density of the energy of the radiation with the fourth power of the wavelength increase and space expansion only accounts for the third power (this is perhaps the biggest problem of Big Bang models, so big that only seldom is mentioned). Note that induction laws can be treated as time-dependent laws and the evanescence of the radiation be directly obtained from the laws; that introduces an unnecessary formal complication. Finally, there are the conservation laws of mechanics, which require a little more attention.
Although it is not usually mentioned, the independence of physical laws in relation to the inertial motion implies the conservation of the weighted mass summation of velocities and square velocities of the particles of a system of a particles. As the A measure of mass is proportional to the weighted mass, these two properties are understood as the conservation of linear momentum and of kinetic energy. Therefore, the correct physical formula of these conservation laws depends on the weighted mass and is valid in both systems of units; for simplicity, the A measure of mass can be used instead of the weighted mass; for instance, for the conservation of square velocity: $$\sum_i{\frac{m_i}{m_{total}}\cdot v_i^2=\text{const}}\Leftrightarrow \frac{1}{2}\sum_i{(m_A)\cdot v_i^2}$$ 
In the first expression the system of units is not indicated because the equation is valid in both systems; in the second equation, velocity can be measured in A or S (has the same value in both) but the measure of mass has to be the A measure. The second expression is the conservation of kinetic energy in A.
The conservation of the angular momentum is the only law modified in A units because the relevant measure of curvature radius is the S measure; the angular momentum L can be written as 
$$\textrm{L}=\mathbf{r}_s\times m_\text{A}\mathbf{v}$$
This is the quantity that holds invariant in an isolated system; the usual A angular momentum, function of $r_A$ 
$$\textrm{L}_\text{A}=\mathbf{r}_\text{A}\times m_\text{A}{v}$$, of an isolated system increases with time:
$$\left ( \frac{d\textrm{L}_\text{A}}{dt_\text{A}} \right )_0=H_0\textrm{L}_0$$ 
This means that the rotation of an isolated rotating body increases with time. For an S observer, this is consequence of the decrease of the size of the body; an A observer can explain this considering that it is consequence of the local expansion of space, that tends to drag the matter.
Note that there is no conflict with the data that support standard physics because this effect is not directly measurable by now; yet, this insignificant alteration has an important consequence: the expansion of planetary orbits.
A note on the value of constants in both systems of units. To change from one system to the other is like any change between two systems of units, but there is a difference: units are time changing one another. As a consequence, a constant in A may not be constant in S. For instance, the Planck constant change in S with the square of the scaling law (a simple image of the physical reason is the following: orbits radii are decreasing in S and also their associated energy; so, the wavelength of emitted radiation is decreasing with orbit radii and also the energy, which implies a Planck constant decreasing with the square of the scaling). Field constants are the exception: they hold constant in both systems of units. The Hubble constant is different: it is constant in S but not in A (in A, the Hubble constant is just the present value of the Hubble parameter). In short, the Hubble constant is the only time constant and is a S constant, field constants are constant is A and S and the other constants, including Planck constant, are relative to atomic phenomena and are constant in A; naturally, the dimensionless relations, like the fine structure constant, are independent of units.
Only the classical fundamental physical laws have been considered because that is what is required, the ground on which the analysis of all the rest of physics can be made, special and general relativity included.
 A: I think something like this really works.* Here's how mainstream physicists would describe it.
Let's use particle-physics natural units ($c=\hbar=1$). Then we have just one dimensionful unit, usually the eV. Nothing is stopping you from using a different unit, the cV (short for "crazy eV"), where the conversion is 1cV = 1eV in this galaxy, but 1cV=0.99eV in the next galaxy over, and 1cV=0.01eV in those very dim red-shifted galaxies etc.
If you start doing physics using exclusively cV's, and you think of cV's as the one true meaningful unit, then you will make statements like "all the electrons and protons are lighter in that galaxy" and "time runs slower in that galaxy" and "distances are shrunk in that galaxy", just like you're saying.*
With a lot of work, it is possible to write the laws of physics in a form where you can directly plug in cV-referenced quantities. But you'll find weird terms. For example, as an electron moves, its mass or energy (in cV) changes, which produces a weird (pseudo)force appearing in the equations. Stuff like that. So you end up with extremely convoluted laws** which are (1) Mathematically equivalent to standard general relativity / quantum field theory / astrophysics / cosmology, yet (2) Much more difficult to think about and work with. This is why nobody does that.*** There are other reasons too.****
FOOTNOTES
*(However I do think you're making some technical errors in your description of exactly how scaling works. The electron charge shouldn't change, because you don't want to change the fine-structure constant. You actually want the forces to scale, not stay fixed, to make it consistent with scaling of photon frequency, I think.)
**The details here are analogous to, or maybe an example of, gauge transformations.
***If you think that inflation, dark energy, and dark matter "do not obey the general laws of the universe, namely the energy conservation principle", then you're wrong. You should spend more time learning about it, or ask a separate question about it.
****Another reason nobody would write the laws this way is, you're proposing that out of all the 100 billion galaxies in the universe, the Milky Way is the galaxy with the heaviest electrons. Are we really so special?? Well, I would say that experience suggests that this kind of assumption is not a promising starting point for scientific theories (cf. Galileo, Darwin, etc.). See also: Cosmological principle.
A: Let us call $f_0$ the frequency at rest and $f$ the shifted one. Then $f_0/f=\sqrt{(1+\beta)/(1-\beta)}=\alpha(v)$, where we called the right side $\alpha$ to simplify. We can rewrite that as $\Delta f/f_0=(1-\alpha)/\alpha=\epsilon(v)$, again, we called the right side $\epsilon$ to simplify.
Now, if you plot the $\Delta f$ vs $f_0$ for H, you will get a straight line with slope $\epsilon$. If you take a different kind of atom, say He, you will also get the same slope, because $\epsilon(v)$ is only a function of $v$.
But if $\Delta f$ is caused by a change in $m_e$, $\epsilon$ will be different for different atomic species (and thus the velocities calculated from them). The reason is that $f_0$ is proportional to $Z^2$, where $Z$ is the atomic number. Thus, for instance, the slope for ionized He will be twice that of H. 
What I do not know is if this effect would be noticeable above noise in the data, my guess is that it is. I am also sure that a change in $m_p$ (due to the constant $m_e/m_p$ ratio) will affect the strong force couplings and here mess up primordial nucleosynthesis, stellar structure, etc, unless you also make changes to other universal constants. 
A: We measure everything, including the atom, with the atom.  There is no external reference. 
The mass unit, in all systems of units is the mass of a determined collection of atoms, put the number.
The mass of the electron is a tiny fraction of  a mass unit, put the number.
Wat is the mass of an atom? It is a tiny fraction of  a mass unit, put the number.  
This is a circular definition because we measure the atom with itself thus, the atom can be of any size. This self-reference is the equivalent to say that it scales whitout the need of any further consideration.   
Try with length unit and find it is linked to atom' radius.
Thus, the atom can be of any size. It is inscribed in our atomic model that the atom can scale. 
I cant find references that demands the absolute size of the atom.  
The best and the accepted way to say that I'm wrong is to present a reference document on the absolute atom's size.  Until then I'will keep saying that the atom scale and there is no Dark Energy, no Dark matter, no BB. No singularities.  
May be that reading the above linked paper you can say what is wrong inside of it, and deny my statements (any reasoning, math, physics).   
On Anna's argument :   

This would be true if the expansion of the universe as postulated by
  the BB model would also expand the atoms and galaxies.

And indeed it does, my model is valid both locally and at large distances, contrary to BB.  The local atoms shrink the same as the lab's ones. And the null measure traces exactlly this fact. And then your argument favours my explanation.
The BB model don't apply locally (GR is not valid in local fields) and the usual argument is that locally the expansion is counterbalanced by the gravitational force. But the expansion is not a force.  
