Hmmm. No, I don't like that either. If the claim was arrived at posteriori then it can be resolved (or at least resolution can be pursued) posteriori.

To return to my Greeks, they may pursue confirmation of their claim by investigating the site of the lighting strike (to make sure it really hit the neighbor's house), or they may check and see whether the neighbor was even home (perhaps Zeus was slaying some visitor). On the other hand, they could investigate the reliability of the individual who informed them that Zeus is responsible for such lightning bolts. Perhaps this individual is mistaken, or fraudulent. They could see if other sources held to be reliable confirm or contradict him. If he relies on some sort of text or artifact they could attempt to test its reliability and ability to produce meaningful data.
In the end, they'll reach the point at which their doubts are satisfied to the extent that the motivation for farther verification no longer outweighs the annoyance of pursuing that verification. They will then be faced with numerous possibilities which they cannot positively reject. They will likely choose the one that seems most likely to them and operate as if it were correct. If they're peasants in ancient Greece, chances are they'll stick with the Zeus theory.

Again, my process in looking into Bob Hope's potential presidency is just the same. I can only try to investigate the reliability of my sources and observations. Perhaps all the history books are wrong about when the US was founded. Perhaps Bob Hope is a time-traveler. Perhaps he is some immortal alien (he always seemed a little odd in a funny kind of way). Eventually I'll reach the point at which my doubts are satisfied to the extent that the motivation for farther verification no longer outweighs the annoyance of pursuing that verification. At that point I'll choose the option that seems to make the most sense to me in light of my experiences. Thus far, I continue to conclude that Bob Hope was not the first US President.

On the other hand, the Greek could have just declared that Zeus was angry with the neighbor and that was all there was to it; it was an unquestionable fact and any evidence for or against it was irrelevant. And I could have declared that it was a fact literally beyond the possibility of dispute that Bob Hope was not the first US President and any evidence for or against that was irrelevant.

The distinction you're talking about has nothing to do with the content of the claim (religious, historical, or whatever) and everything to do with how that claim was formulated. If the claim was formulated a posteriori then its resolution can be pursued, at least to some degree, a posteriori. If the claim was arrived at a priori then it cannot be pursued a posteriori at all. But eventually all claims end up resting on a priori assertions.