Question
Section 1.5 on page 107 of the full SI brochure, Le Système international d’unités, 8ᵉ édition, comments on physically realizing unit definitions when accounting for relativistic effects (I personally have emboldened the first sentence):
The definitions of the base units of the SI were adopted in a context that takes no account of relativistic effects. When such account is taken, it is clear that the definitions apply only in a small spatial domain sharing the motion of the standards that realize them. These units are known as proper units; they are realized from local experiments in which the relativistic effects that need to be taken into account are those of special relativity. The constants of physics are local quantities with their values expressed in proper units.
Physical realizations of the definition of a unit are usually compared locally. For frequency standards, however, it is possible to make such comparisons at a distance by means of electromagnetic signals. To interpret the results the theory of general relativity is required since it predicts, among other things, a relative frequency shift between standards of about $1$ part in $10^{16}$ per metre of altitude difference at the surface of the Earth. Effects of this magnitude cannot be neglected when comparing the best frequency standards.
(If you need a refresher of the explicit definitions of the base units, scroll down to the Definitions header.)
This question in particular pertains to the metrology, not the relativity:
In practice, what are some ways in which a scientist could realize the definitions of units within an environment subject to relativistic transformation such that the units could be used to indicate those transformations?
Definitions
Unless you have them memorized, you may find it helpful to reference the following definitions of the seven base units as quoted from section 2.1.1 of the aforementioned full SI brochure, although original stylization is not reflected:
The metre is the length of the path travelled by light in vacuum during a time interval of $1/299 \, 792 \, 458$ of a second.
It follows that the speed of light in vacuum is exactly $299 \, 792 \, 458$ metres per second, $c_0 = 299 \, 792 \, 458 \ \left.\mathrm{ m }\middle/\mathrm{ s }\right.$. (p. 112)
The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram. . . .
It follows that the mass of the international prototype of the kilogram is always $1$ kilogram exactly, $m(\mathcal{K}) = 1 \ \mathrm{ kg }$. (p. 112)
The second is the duration of $9 \, 192 \, 631 \, 770$ periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.
It follows that the hyperfine splitting in the ground state of the caesium 133 atom is exactly $9 \, 192 \, 631 \, 770$ hertz, $\nu ( \mathrm{ hfs \ Cs} ) = 9 \, 192 \, 631 \, 770 \ \mathrm{ Hz }$. (p. 113)
The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed $1$ metre apart in vacuum, would produce between these conductors a force equal to $2 \times 10^{−7}$ newton per metre length.
It follows that the magnetic constant, $\mu_0$, also known as the permeability of free space, is exactly $4\pi \times 10^{−7}$ henries per meter, $\mu_0 = 4\pi \times 10^{−7} \ \left.\mathrm{ H }\middle/\mathrm{ m }\right.$. (p. 113)
The kelvin, unit of thermodynamic temperature, is the fraction $1/273.16$ of the thermodynamic temperature of the triple point of water.
It follows that the thermodynamic temperature of the triple point of water is exactly $273.16$ kelvins, $T_{\mathrm{tpw}} = 273.16 \ \mathrm{ K }$. (p. 114)
The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in $0.012$ kilogram of carbon 12. . . .
It follows that the molar mass of carbon 12 is exactly 12 grams per mole, $M(^{12}\textrm{C}) = 12 \ \left.\mathrm{ g }\middle/\mathrm{ mol }\right.$. (p. 115)
The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540 \times 10^{12}$ hertz and that has a radiant intensity in that direction of $1/683$ watt per steradian.
It follows that the spectral luminous efficacy for monochromatic radiation of frequency of $540 \times 10^{12} \ \mathrm{ Hz }$ is exactly $683$ lumens per watt, $K = 683 \ \left.\mathrm{ lm }\middle/\mathrm{ W }\right. = 683 \ \left.\mathrm{ cd \ sr }\middle/\mathrm{ W }\right.$. (p. 116)
It may be worth noting that the definitions of the kilogram, second, kelvin, and mole have relatively minute specifications about (in respective order) the contamination, thermodynamic temperature, isotopic composition, and the atomic bonding and energy level of the matter to which they refer.