I should have realized this sooner: P(B3|~A3) is just the updated value of r, which isn't affected at all by (A1,B1,A2,B2). So of course the answer according to this model should be 1/3, as it's the expected value of r in the prior distribution.

Still, it was a good exercise to actually work out a Bayesian update on a continuous prior. I suggest everyone try it for themselves at least once!