"I disagree; I think the underspecification is a more serious issue than the uncomputability. There are constant factors that outweigh, by a massive margin, all evidence ever collected by our species."

Agreed. The constant factors really are a problem. If one has taken a few information theory courses, it's easy to disregard it as one usually uses Kolmogorov on e.g. symbol sequences in the infinite limit. When comparing two theories though, they have finite size and thus constants does matter. It is probably possible to find two Turing machines such that two competing models have equal length on their respective best machines even if they differ greatly when tested on one of them.

It may be possible to construct an argument that favors an interpreter over all others, Sebastian Hagen gave a few ideas above, but it is highly non-trivial.