The Red Shift is not an assumption.
What you are discribing has nothing to do with the "speed" of light, it has to do with its frequency, or wave-length, as seen by a distant observer. When a distant object is moving towards the observer at a significant percentage of C its wavelength is compressed, or shortened, appearing blue to the observer. As a distant object moves away from the observer its wavelength is expanded, or lenghtened, appearing red.
This shifting towards red or blue would happen regardless of the measured speed of C. Since E=MC2 works well enough (to as many decimal points as we can measure) that Hiroshima was vaporized, I respecfully suggest that Einstein was closer to the correct answer than any fanciful idea that C is not a constant in a vacuum.
Remember Occam's Razor: A rule in science and philosophy stating that entities should not be multiplied needlessly. This rule is interpreted to mean that the simplest of two or more competing theories is preferable and that an explanation for unknown phenomena should first be attempted in terms of what is already known.
The cosmological red shift assumption is indeed in question. The assumption is that the more distant the light source the faster it is receding from us. Could something else than mere velocity cause the observed shift in spectra? Also, this article throws a monkey wrench into the expansion of the universe model since it means the orderly distance measurements and therefore the timeline would be thrown off.