As much as I hate to say it, I think Hamilton's influence in the election of 1800 is somewhat over rated. He did write some letters to his friends expressing his utter contempt for Burr but I do not think he was responsible for the "deal" that finally broke the stalemate.
As I understand it, a few moderate Federalists tried to get Jefferson to commit to not dismantling the Navy, not firing all of the Federalists holding Federalist office and keeping the Hamilton financial system in place.
Jefferson refused to be pinned down but several of his friends made vague sounding comments about Jefferson's intentions and that was good enough for Congressman Bayard of Delaware who abstained on the last ballot, thus giving Jefferson a majority of the states voting.
The country was pretty fed up with the stalemate, the governor of Virginia (I believe it was Monroe) was threatening to march the Virginia militia to Washington if the "people's choice" was not ratified and Burr was silent thoughout the entire process.
A last ditch Federalist scheme to elect a temporary President, probably Secretary of State John Marshall, died still born and the inevitable ratification of Jefferson ocurred. For a much more detailed description of the entire mess and its implications for the future, I suggest "The Failure of the Founding Fathers, Jefferson, Marshall and the Rise of Presidential Democracy" by Bruce Ackerman.
If it had been different and Burr had been elected, I thonk he would have been a minor one term President. The Jeffersonians would have opposed him at every step and the Federalists would have never trusted him. I do not think that Burr was the evil genius of American history who would have created a Burr imperial dynasty. He was a charming political fixer who showed in OTL a real ability to get into trouble and then to squirm out of it.
AH