Well, perhaps the Avengers are summoned to the Hamilton - era by some magical or time - rift related incident. Black Widow could blend in easily and start gathering intelligence on the political situation. Thor might be mistaken for a god, which could either cause chaos or inspire the people. And Hamilton, being the ambitious man he is, might see an opportunity to use the Avengers to strengthen his own position, leading to some interesting power dynamics and moral dilemmas within the fanfiction.
In this fanfiction, it could be that the events of the Avengers' battles have somehow disrupted the time - space continuum and they end up in the Hamilton universe. Spider - Man, if he's part of the Avengers in this scenario, could use his web - slinging abilities to move around the city in the Hamilton - era in a really cool way. Hamilton, on the other hand, might start to question his own beliefs when faced with the superheroes from another world. The Avengers would have to deal with the cultural differences and also find a way to get back to their own time while not causing too much disruption to the Hamilton - era events.