To make this meaningful, we must add a probability distribution over mathematical structures, effectively assigning each a degree of existence.

Probability doesn't work where indexical uncertainty is at play. You are not allowed to assign probabilities to situations and act as if you are working with one of the possibilities, when it's possible that you are actually orchestrating the outcome from multiple locations at once. Two worlds, for example, could be simulating each other, which makes any independence assumptions go away.