Why is this relevant? Because if for example (and no, I'm not saying this is what happened), God came out of the sky, pointed at me, and said "I exist." I would know that either he existed, or something else did that was trying to fool me into thinking he did. Either way I would have belief that something supernatural (outside of the realm of what human science commonly accepts) had happened.

Actually, my default response for this sort of thing is to immediately go to a hospital, and get a head CT and a chat with a certified psychiatrist. I mean, sure, it could be the supernatural, but we KNOW mental illness happens. The priors for me being crazy (especially given some unique family history) are not very low. Much, much higher than the odds of a deity actually existing, given the aforementioned Occamian priors.

How do you prove that a godlike entity exists if it doesn't want to be proven? Assume that it has complete freedom to move through time so that tricking it doesn't work because it can just go back in time (that's what omnipotent means after all). And that you don't know the reasons why it's staying hidden so no argument to try to get it to show itself will work.

You don't. Rationalism only works if God isn't fucking with you. That said, there's a huge space of possible constructs like that one (entities that conveniently eliminate all evidence for themselves). It's not infinite, but it's arbitrarily large. From a rationalist's perspective, if any of them were real, we wouldn't know, but the odds of them actually being real in the first place are... not high. Again with the Occamian prior. So, I'm not much moved by your analysis.

That said, I am curious what your personal experience was.