Mightn't the vet have already factored the base rate in? Suppose x is the more common disease, but y is more strongly indicated by the diagnostics. In such a case it seems like the vet could be justified in saying that she cannot tell which diagnosis is accurate. For you to then infer that the dog most likely has x just because x is the more common disease would be putting undue weight on the Bayesian priors.
If it's worth saying, but not worth its own post, then it goes here.
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