How would one tell if the universe uses the real value Pi or a finite precision version of Pi whose finiteness is epsilon greater then what is needed to calculate any observable value?

You can't. However, if you somehow found an encoding of a physical constant that was highly compressible, such as 1.379[50 digits]0000000000000000, or some other sort of highly regular series, it would be strong evidence towards our universe being both computable and, indeed, computed. (No such constant has yet been found, but we haven't looked very hard yet)