You can take any equation that has "c" as a component and consider the implications as c approaches infinity as a limit. For example "E=MC2" implies that in a universe with an infinite speed of light you could extract infinite energy from a finite amount of mass, and the characters in the story could discuss this "known fact", which powers their space ship.
It wouldn't make sense to do that, since you wouldn't have that equation in a universe where c is not finite. You probably wouldn't have people either, come to think of it.
It's not like the equations of physics exist by themselves, like some platonic ideals. They represent something about the workings of the world. The finite speed of light pops up in Maxwell's equations, so it says something about the workings of everything involving EM interactions, from propagation of signals to chemistry. That there is a universal speed limit is also telling us something about cause and effect relationships between events in space-time, and about the evolution of the entire universe.
These are fundamental rules that make the world what it is and getting rid of them is a sure way to unmake it completely.
I actually did think of that, but remember that Special Relativity was completely developed as a thought-experiment. The equations come out of the assumptions. Someone in a universe where it's
wrong could still do the thought experiment and come up with the equations. So, you get E=MC2 if you
assume a finite speed of light that's invariant. Someone in the story could laugh at whoever came up with that idea that's completely wrong in their universe, but correct in ours. For example, their version of Maxwell's equations could have proven that the speed of light is
invariant in all frames of reference, and their version of Einstein could have worked out Special Relativity from that. But then the scientific debate would have been whether the speed of light is infinite (instantaneous) or finite, with the consequences that the equation E=MC
2, which is true in both universes implies you can build perpetual motion machines or somesuch in their universe. That aspect of the story could revolve around them discussing that since the speed of light was first proven to be
invariant then naturally it all made sense when it was found to be infinite, and how silly the universe would be if the "finite speed of light, yet somehow invariant to all observers" people had their way. I mean, it wouldn't be hard to write a non-relativity universe story where they say the same thing you are saying, and say that a universe where relativity is
true is to silly for words and how could any living beings possibly live in a universe where time and space were so distorted.