Lowercase p(B|A) seems to really mean p_B(h(a)), and that's not obvious. Hell, I might still have it wrong.I think you have it about right. It is "semi-obvious" by the fact that of course, observation and state are related through the observation function (in your case, observing a 3D point as a 2D coordinate) and that is of course part of the PDF.
And most everyone says to ignore p(B) in the denominator, but that's really sloppy hand waiving.
It's not. You want to know the most probable value of A. In other words you are looking for the argmax of a function of A. p(B) is purely a function of B, no A involved, so it becomes a constant in your equation. Since it's in the denominator, it's a normalizing constant.
Note that if you have the "Uppercase Bayes" (P(A|B) = ...) you are looking for concrete values so the normalizing P(B) does matter.
Now in the case of "lowercase Bayes" (p(A|B) = ...) it matters just as much, but you can still ignore it if all you're looking for is the argmax of the resulting PDF, as the p(B) is just a scaling constant and it's not changing the argmax of p(A|B).
but in each of the four terms, they do something radically different. I can't see a pattern to follow.
I don't understand what you mean here.