As a property of the coin and the flip and the environment and the laws of physics, the probability of heads is either 0 or 1. Just because you haven't computed it doesn't mean the answer becomes a superposition of what you might compute, or something.

What you want is something like the result of taking a natural generalization of the exact situation - if the universe is continuous and the system is chaotic enough "round to some precision" works - and then computing the answer in this parameterized space of situations, and then averaging over the parameter.

The problem is that "natural generalization" is pretty hard to define.