Does a precise definition of the meter not involving light exist, so that variation of $c$ can be tested? The second is defined as 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 cesium 133 atom.
The metre is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second.
With those definitions, we are left with no way to check the invariance of $c$ (the local speed of light in freely falling and non-rotating reference frames). Is there a definition of length which would be independent on anything involving light, and allow to check the invariance of $c$?
I am aware of classical definitions based on pendulum or earth's meridian. I red a definition involving the wavelength of a certain type of radiation, but this again depends on the speed of light. I am wondering it there exists a modern alternative definition of the meter, light independent and usable for precise experimenting.
 A: The issue isn't whether we have a definition of the SI base units involving light. The issue is that it is never possible to define empirically whether or not a universal constant is in fact non-constant, except in the case where it's unitless. See Why do universal constants have the values they do? .
dmckee comments:

You don't need to know the length of a meter to do the proposed comparison. You just need to cart the apparatus used in one environment to the other environment. That makes the length you use "1'experimental pathlength" in both cases. Problem solved.

But this doesn't accomplish what the OP wants. The OP wants to test whether or not the speed of light is the same everywhere.  Say you have a totally self-contained apparatus that measures $c$ and outputs its result as a number. You move the apparatus somewhere else, and it outputs a different result. One possible explanation is that $c$ varies from point to point. Another possible explanation is that some other universal "constant" has changed. For example, Planck's constant $h$ could have changed. This would change the sizes of atoms in the apparatus, as well as having other effects. We can never say whether $h$ really changed or $c$ really changed. What we would be able to test is whether the fine structure constant changed, because the fine structure constant is unitless.
