How do you decisively refute someone who claims that if only we had a better theory, we could calculate the probabilities?

This seems like too strong a statement. After all, if one knows exactly the initial quantum state at the Big Bang, then one also knows all the freebits. I believe that what you are after is not proving that no theory would allow us to calculate the probabilities, but rather that our current best theory does not. In your example, that knowing any amount of macrofacts from the past still would not allow us to calculate the probabilities of some future macrofacts. My question was about a potential experimental signature of such a situation.

I suspect that this would be a rather worthwhile question to seriously think about, potentially leading to Bell-style insights. I wonder what could be a simple toy model of a situation like that: a general theory G, a partial theory P and a set of experimental data E from which one can conclude that there is no well calibrated set of probabilities P->p(E) derivable from P only, even though there is one from G, G->p(E). Hmm, I might be letting myself to get carried away a bit.