You could also look at it the other way: Using the same doctor and lab and procedure is the best way to eliminate a false positive, because if the cause was external, then the cause may not be repeated. But if you went to a new lab/doctor/whatever, you've now introduced new variables that could cause a false positive on top of whatever already caused it.
Given that it could go either way, it makes sense to think of each one as independent.
Now if you really wanted to take advantage of Bayes, if you got a positive test then you should get two more tests, one with the same lab and one with a totally independent lab (or even if you got a negative test, assuming your first Bayes run gives a 50% confidence)
More information allows for better reasoning. Repeating the same test and also doing a "more independent" test are the minimum next two things you should do given a positive result on a single test.
Given that it could go either way, it makes sense to think of each one as independent.
Now if you really wanted to take advantage of Bayes, if you got a positive test then you should get two more tests, one with the same lab and one with a totally independent lab (or even if you got a negative test, assuming your first Bayes run gives a 50% confidence)