The basic assumption of SR is actually both simpler and more general than the one you have quoted: namely, that the laws of physics are the same in all inertial frames. Practically speaking, this means that there is no local experiment which you can perform in order to distinguish between two given inertial frames - the necessary and sufficient condition for this is that the causal structure of these two local spacetime patches must be identical.
The easiest way to experimentally probe the (classical) causal structure of local spacetime is to set up a source of radiation in vacuum, and examine how this radition propagates through space and time. So let us consider some source-free radiation field in vacuum; in any arbitrary inertial frame, this field will obey the general homogenous wave equation
[math]\square f(x,t)=0[/math]
with some function f(x,t).
SR is now telling us that the same is true in all other inertial frames as well, meaning that no matter which inertial frame you look at, the local radiation field will always obey the above wave equation. This necessarily means that c must be invariant between inertial frames. This is something we can of course test, which has been done innumerable times with many different setups, both directly and indirectly.
No. But what we can do is look for clues as to whether the fundamental constants have changed over time, or not. One way to do this is to look at the phenomenon of natural nuclear fission reactors, such as the one in Oklo:
https://en.m.wikipedia.org/wiki/Natural_nuclear_fission_reactor#Relation_to_the_atomic_fine-structure_constant
Since the fine structure constant directly depends on c (and vice versa), this would seem to indicate that - at least in the past 2 billion years or so - there has been no change in the numerical value of c.