Are increases in Planck's constant and the speed of light operationally distinguishable? If only dimensionless constants are physically meaningful, and both Planck's constant, $h$, and the speed of light, $c$, are in the denominator of the expression for the fine structure constant, $\alpha$, then is there any way of distinguishing, by measurements or experiments, between an increase of $h$ and an increase of $c$ (assuming a definition of $c$ in terms of a meter rod, rather than itself)?
 A: What Duff says makes sense. Only a change in a dimensionless combination of units can be detected. 
The argument seems to be that every physical constant (unit) must be defined and measured in terms of others. So we cannot tell if a change is in the unit being tested or one of the units which we assume to be constant. Which set of units could have changed depends on how the unit in question is defined.
A: This is an experimentalist's answer:
What is the fine structure constant?


A dimensionless constant which characterizes the electromagnetic force
The  coupling constant is also called the "fine structure constant" since it shows up in the description of the fine structure of atomic spectra. It appears naturally in the equations for many electromagnetic phenomena.

Alternatively, there are other equivalent forms:


e is the elementary charge;
π is the irrational number Pi;
ħ = h/2π is the reduced Planck constant;
c is the speed of light in vacuum;
ε0 is the electric constant or permittivity of free space;
µ0 is the magnetic constant or permeability of free space;
ke is the Coulomb constant;
RK is the von Klitzing constant;
Z0 is the vacuum impedance or impedance of free space.

It is known that the coupling constant  changes with the energy of the interactions:


Figure 5: Evolution of the inverse of the three coupling cons
tants in the Standard Model (left) and in the supersymmetric extension of the SM (MSSM) (right). Only in the latter case unification is obtained. The SUSY particles are assumed to contribute only above the effective SUSY scale MSUSY
of about 1 TeV, which causes a change in the slope in the evolution of couplings. The thickness of the lines represents the error in the coupling constants

The a_1 is the electromagnetic constant in this plot, and has been measured to change, i.e. an experimental fact.
The question then is, for the measured change in the dimensionless by construction number, can we also find which of the dimensionfull constants is contributing to the change?  The answer is :an independent experiment is needed , devised to measure any changes in the dimensionfull numbers individually. Something must be changing, the individual constants have been found by independent measurements at low energies.
There is no algebraic way that from one measurement, two values can be deduced, i.e. from a change in alpha, a unique change in one of the various dimensionfull constants. One needs two equations for two unknowns and here there is only one.
I would be in favor of the change being in the permittivity or the permeability of free space, as quantum field theory populates it with various fields, but in any case, an independent  measurement is necessary.
