Counterexample: I have a Platonic view of mathematical truths, but a Bayesian view of probability.

That's interesting. I had imagined that people would be one way or the other about everything. Can anyone else provide datapoints on whether they are Platonic about only a subset of things?

... in order to triangulate closer to whether Platonism is "hard-wired", do you find it possible to be non-Platonic about mathematical truths? Can someone who is non-Platonic think about them Platonically -- is it a choice?

For any given coin flip, either the fundamental truth is that the coin will come up heads, or the fundamental truth is that the coin will come up tails. The 50% probability represents my uncertainty about the fundamental truth, which is not a property of the coin.

See, that's just not the way a frequentist sees it. At first I notice, you are defining "fundamental truth" as what will actually happen in the next coin flip. In contrast, it is more natural to me to think of the "fundamental truth" as being what the probability of heads is, as a property of the coin and the flip, since the outcome isn't determined yet. But that's just asking different questions. So if the question is, what is the truth about the outcome of the next flip, we are talking about empirical reality (an experiment) and my perspective will be more Bayesian.