It may be just a point of view
Speed of light is our reference since the formulas make out to be simpler that way. It is definitely possible to set up formulas where a changing light speed over history is used.
To explain big bang and inflation we could see it as light speed dropping while the universe stay att the same size. Inside the universe a reduction of light speed would look exactly same as inflation since that is what we use to measure distances.
For me that way of looking at it is useful to understand how big bang could happen everywhere. It did not happen in a single point from which it expanded.
Since It only changes the point of view, I can not imagine any experiments that could prove it one way or the other. What could happen though is that with more observations more complicated formulas are needed to explain them and some day using a variable light speed would make a less complicated model.
For now Ochams razor say light speed is constant.