The way I see it, there are no independent constants in the universe. They are all emergent values; all related. There was no possibility of one being different while the others remain the same resulting in a failed/lifeless universe. Change one constant and all the others would change proportionally with the result that there is no discernible difference, hence no alternate universe is possible. So i'd say our set of constants are just one set of a possible one set. :)
Now to answer the OP...
Einstein was wrong but not because of the big bang, as that theory is wrong too. The idea that the speed of light is a universal speed limit and not source-dependent is a falsity that was proven wrong back in the sixties when powerful radar signals were bounced off the planet Venus from multiple radar stations around the globe simultaneously. As Bryan G. Wallace showed, signals from the side of the earth rotating towards Venus came back sooner than those from the side rotating away, to a degree that fits source-dependent models:
http://www.ritz-btr.narod.ru/wallace.pdf
This suggests that [as common sense would have it] the speed of light is additive, i.e. light emitted from a source with velocity v, will have a velocity of c+v on emission. Light is capable of travelling through space with differing velocities. Wallace's observation refutes relativity but unfortunately, by this time relativity had already become more like a religion than science. No career-minded physicist could be seen to be paying heed to a "relativity-denier", so Wallace's paper did not get the attention it deserved and no further investigation was made. The mainstream scientific community carried on believing in the fallacy that the speed of light is always constant.
Experiments on earth can give the illusion of source-independence but that is due to the EM fields of the earth's matter interfering by normalizing the speed of light from a moving source, i.e. slowing it down to c almost as soon as it is emitted. Astronomical and interplanetary observations reveal light's base nature. It's planet-bound behaviour is a special case.
There have been other astronomical observations that suggest source-dependence, like this:
http://www.news.ucdavis.edu/search/news_detail.lasso?id=8364
But such observations are seen as some minor curiosity and the theorists will bend over backwards to invent any kind of fantastical nonsense to explain them so that they can avoid ditching relativity.
The time-dilation seen in type 1a supernova is likely down to light being source-dependent too (rather than expanding spacetime!). And, many apparently variable stars may well be binaries in which light bunching is occurring as faster light emitted from a star moving at it's maximum velocity towards us in it's orbit, catches up with slower light it emitted later:
http://www.datasync.com/~rsf1/binaries.htm