Point of view is a literary concept, so the tense would primarily depend on how much your narrator knows. In addition, character's speech, aloud or internal, would most naturally defer to the present.
the claim "Washington is the president" is an embedded noun clause, the only conjugation that you need to worry about is the main clause, since within the subordinate clause the different verb tenses don't create an unintelligible sentence. "Washington was/is/will be president" does not effect "a person who lived could claim" which is the main clause.
"could" is an auxiliary verb modifying the verb "claim" and wouldn't cause the conjugation of the verb in the embedded noun phrase.
Another issue, is the inclusion of "that" which is, perhaps, the biggest problem. "that" can be a number of parts of speech which then change the meaning of the phrase, in your particular example "that" is causing most of the ambiguity. Why mess around with a noun clauses if you don't have to?
example: "As i've been narrating this story from the beginning, and you've seen that i've gotten a few things wrong, you wouldn't be surprised that someone from this story would say he could truthfully claim that, "washington is the president" but still be untrue.
or with the exclusion of "that"
A person who lived in 1777 could claim Washington is president.