It is not clear what a variation in a dimensionful quantity actually means, since any such quantity can be changed merely by changing one's choice of units. John Barrow wrote:
"[An] important lesson we learn from the way that pure numbers like á define the world is what it really means for worlds to be different. The pure number we call the fine structure constant and denote by á is a combination of the electron charge, e, the speed of light, c, and Planck's constant, h. At first we might be tempted to think that a world in which the speed of light was slower would be a different world. But this would be a mistake. If c, h, and e were all changed so that the values they have in metric (or any other) units were different when we looked them up in our tables of physical constants, but the value of á remained the same, this new world would be observationally indistinguishable from our world. The only thing that counts in the definition of worlds are the values of the dimensionless constants of Nature. If all masses were doubled in value [ including the Planck mass mP ] you cannot tell because all the pure numbers defined by the ratios of any pair of masses are unchanged."[5]
There available evidence seems to rule out any *observable* spacetime variability in fine structure constant. Any variability in the fine structure constant would seem to be much less than variations in the Hubble Constant, so a "varying speed of light" seems to be ruled out as an explanation.
There are also some interesting secondary effects, such as much higher energy releases in distant/old processes because c^2 is much greater (quasars?) and a net loss of energy over time which might account for the time's arrow problem.
But I'm almost perfectly ignorant here.