Okay, everybody, I'm seeing the 'Burr cut short a great career' myth here, so I'm going to do what I can to put it to sleep. Hamilton was, by the time he caught that bullet, one of the great political burnouts. The Federalists themselves hated him for sabotaging his own party's Presidential candidate in the last election--he'd made enemies all over Washington, both from his double-dealings, and his abusive manner--and he was still damaged from a scandal he'd survived, by copping to an affair. Simply put, if Hamilton had even wanted to run in 1808, there was no way in Hell the Federalists would nominate him. The change you want in Hamilton to save the Federalists isn't "don't have him get shot", it's "don't let him be an egotistical jerk who feels that his intelligence gives him a right to insult people around him, and is willing to pit his party against itself."
And at that point, is he really still Hamilton?