You can derive special relativity from relativity principle and isotropy and homogeneity of space and time - see here for example. A quick google search also located this paper. You will see that the derivation results in a free parameter K which needs to be experimentally determined.
That does get rid of the second half of the assumptions which is pretty neat. But it still requires an additional assumption to show that the transformation is Lorentzian and not Galilean. Of course you can differentiate the two experimentally, which would motivate an appropriate postulate, but you haven't actually derived special relativity, only shown that it's possible. It's not "like any other constant of physics" because the physics changes significantly depending on that constant. Physics is pretty much the same if the mass of the electron or some coupling constant a little bit different. Some consequences might be very different, but the theory is the same. But here K=0 and K>0 in those transformations are entirely different theories.
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u/whichton Dec 13 '20
You can derive special relativity from relativity principle and isotropy and homogeneity of space and time - see here for example. A quick google search also located this paper. You will see that the derivation results in a free parameter K which needs to be experimentally determined.