There are an infinite number of true provable theorems in PA and an infinite number of true unprovable theorems. So any measure must go to zero for all but a finite number of sentences I think there will be a problem with knowing p in that one can search all the lower weighted sentences and find the percentage of true sentences so that a point may be reached where some sentences may have to be undecidable to make the probability work. If this is the case then then knowledge of the probability would make some sentences decidable so the probability can't be known and the proof would have to be non-constructive that it exists which could cause problems with some philosophies of mathematics like intuitionism. I recall a similar question about quantum computers in which the answer was 1/2.
I have found a reference that says that no halting probability is computable. If proof could be made into a process that either succeeds or never halts then that would provide evidence that the probability of an undecidable theorem is uncomputable