I think the pull of truth is what motivates Norman's introspection: what he ultimately desires is a true belief, not one which is coherent. At each step, he assesses the evidence in light of that goal, and constructs the belief that is the most reasonable approximation he can make of the truth.

Irrespective of knowledge, it feels correct to say that Norman had a reasonable belief the the president was in Florida until he got more evidence. It also makes sense for Norman to characterize his beliefs as reasonable without the need to invoke an outside observer. That belief is grounded in truth as a goal, but ultimately independent of the actual facts in the matter.

Suppose we say that Norman is actually a brain in a vat, and that the president was a figment constructed by alien epistemologists experimenting on his perception. This doesn't and can't change his beliefs since it doesn't alter his evidence: his beliefs are still reasonable since they're the best approximation of truth he's capable of constructing. Norman can never say for sure if any evidence he gets is actually indicative of the truth, but he's still capable of engaging rationally with his evidence in an attempt to seek it out.