I might be misreading you...still I'm just going to point out here that changing a constant to a different constant is not anywhere near what you have described. What you've talked about here is a misapplication of math, and is just plain silly.
C is for information propagation, not light; it's used that way in laymans terms because it's easier to associate it as light speed for a layman trying to understand why a lightswitch "instantly" brightens a room.
EG, If you substitute a different value in another constant equation, say your savings account's interest rate, do you think you'll get the same results as with the actual APR? The same is true for other things like shear modulus; If you substitute the vale of steel with the value of Styrofoam, do you think the results will be the same?
Well, The problem is that the concept behind the equation, the justification as to why they chose to use the measurements and constant they did, that concept SHOULD still work as a theory if you use another constant. The reason is simply because the PURPOSE of the constant of light speed, has nothing to do with the numerical value of 300,000,000 . Its simply to create the exact same level playing field for both observers, having already assumed that both observers must be on different pages with their measurements in the first place. In this scenerio, unlike an equation about my bank balance, we should be able to use some other units system for the velocity of light, maybe imperial, 186,000, or just call light speed ONE unit. As long as its the same value for both observers. It only provides the single common "yardstick" for both observers, nothing more.
Have a look through the entire derivation of LT or SR that illustrates the process using the comparison between stationary and moving observers, watching the motion of a photon as the light clock moves sideways. One sees a pure vertical path, the other sees a zig zag path.
Nowhere in this setup is it critical that the thing doing the zig-zag be a photon. It was chosen because it was claimed to be the best and only thing that goes the same speed as measured in any frame. (a claim that is unprovable, and illogical)
So even if I agree that light is the only thing that goes the same speed for all, Its still totally possible that we can use the constant speed of sound, as long as the moving guy KNOWS that he is moving, he can make an adjustment to his reading for the speed of sound, compensating for his own motion.
But if you do this, you can't get the gamma result. Why? well because the idea that the photon or sound wave front could possibly be doing two totally different things just because two observers are watching from different vantage points, is INSANE. One action of a physics object CAN NEVER equate to TWO contradicting measurements UNLESS one of both observers are ignorant of some critical fact. In SR and LT the moving observer is PROHIBITED from knowing his true condition, that he is moving and MUST take that fact into consideration in his math. GIGO.. garbage in, garbage out. That's the summary of LT and SR.
The root cause for the insanity of relativity is this idea that an imaginary reference frame can actually make some difference to real physical objects, and even the concept of time, this is insanity. I cant put it any other way. Its just crazy to entertain that idea and still want to do Physics.
And it does not matter one jolt if you find it important to get all anal about the idea that the speed of light is not the "speed of light", rather, its the "speed of causality".
c is roughly 299,792,458 meters per sec. This is an AGREED vale not a measured value. Anyway, the jury is still out as to whether light "travels" at all!
As I said many times, in Physics we don't understand very much about light at all. Currently we have 3 claims as to how light works, its a WAVE, its a Particle, its spaghetti. (the last one is my idea) All three can be forced to work using the gooey stuff called Mathematics.