but there's always a computable number with less than epsilon error, so would this ever actually matter?

Yes. There are programs that can neither be proven to halt nor to not halt. The knowledge that these do not halt would need to be specifically programmed in, and the K-complexity would increase for each one.

On the other hand, it's not exactly known how common these are. They might be pretty rare.

In any case, you could change it from shortest computer program to shortest definition, and thus get definable numbers instead of computable ones.