You can as well argue in the same way you argued about the meter with the fine structure contant since it also
Nugatory said:
You are right that a change in ##c## need not imply a change in ##\alpha##, but your fellow forum members are making a weaker claim: When someone asks about the physical consequences of the speed of light being different, we should instead be talking about a change the value of ##\alpha##.
I think one can make the point clear from this example, using the new SI. In fact the new SI is the (almost) most transparent definition of a coherent set of (base) units we have, given our current fundamental natural laws. Of course we need the fundamental natural laws as far as we know them to define our units.
The new SI is based on a set of general fundamental constants, except the second for practical reasons, i.e., because we still cannot determine the value of the gravitational constant given the present definition of the base units to also use its value as defining the SI units completely with fundamental constants.
That's why the SI still uses ##\Delta \nu_{\text{Cs}}## to define the second, i.e., the frequency of the em. wave emitted due to the groundstate hyperfine transition of Cs-133:
The second, symbol s, is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency, ##\Delta \nu_{\text{Cs}}##, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to ##\text{s}^{−1}##.
All the other (physical) base units define the values "fundamental constants of Nature", according to our current understanding of these laws. So, indeed, the metre is defined by just choosing a value for the limiting speed of relativity, which empirically is to a very high accurcy the phase velocity of electromagnetic waves in a vacuum:
The metre, symbol m, is the SI unit of length. It is defined by taking the fixed numerical value of the speed of light in vacuum ##c## to be 299792458 when expressed in the unit ##\text{m} \cdot \text{s}^{-1}##, where the second is defined in terms of the caesium frequency ##\Delta \nu_{\text{Cs}}##.
Then to define the kg the Planck unit of action (not the modified Planck constant!) is used:
The kilogram, symbol kg, is the SI unit of mass. It is defined by taking the fixed numerical value of the Planck constant ##h## to be ##6.62607015 \cdot 10^{−34}## when expressed in the unit J⋅s, which is equal to ##\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-1}##, where the metre and the second are defined in terms of ##c## and ##\Delta \nu_{\text{Cs}}##.
Finally for this argument we need the definition of the unit of electric charge or, equivalently, of the electric current:
The ampere, symbol A, is the SI unit of electric current. It is defined by taking the fixed numerical value of the elementary charge e to be ##1.602176634 \cdot 10^{-19}## when expressed in the unit C, which is equal to A⋅s, where the second is defined in terms of ##\Delta \nu_{\text{Cs}}##.
Now we have the four base units defined needed for the argument, why it makes sense to ask and to decide empirically, whether the fine structure contant has changed over time. The finestructure constant is a dimensionless quantity defined by
$$\alpha=\frac{e^2}{4 \pi \epsilon_0 \hbar c}.$$
Here everything has defined values, except ##\epsilon_0##, which must be measured, given the values of ##\hbar=h/(2 \pi)##, ##c##, and ##e##, which are all defined values when expressed in the SI units according to the above quoted 2019 definition of the SI base units (s, m, kg, and A).
The current state of the art is ##k_e=1/(4 \pi \epsilon_0)= 8.9875517923(14) \cdot 10^9 \text{N} \cdot \text{m}^3 \cdot \text{s}^{-4} \cdot \text{A}^2##. It's determined (according to the CODATA-2018 paper) by measuring the anomalous magnetic moment of the electron or recoils of atoms when emitting em. radiation.
