Since our resident AHer Alexander Hamilton is unavailable. I will attempt to step into his shoes.
To answer the question yes he could have been. One thing to note, he never held an elected office. He was Treasury Sec. that was until 1796. and a Maj. Gen. from 97-98, during the New Army. How likely was it after the Election of 1800? Not very, He was considered by many after 1800 to be untrustworthy and a manipulator because of his pamphlet published against Adams in the 1800 election I don't remember the exact title but its was something like "A. Hamilton's indictment of the public and private character of John Adams". After that pamphlet his party started to fall apart.
Butterfly that pamphlet and he has a shot. Butterfly the 1794(?) Mariah Reynolds affair and it is more likely.
Really without any significant butterflies 1796 is in my opinion his best shot. J. Adams was never the most popular guy, especially if you get Washington to endorse his protege.
One other POD to consider that hasn't really been done (although I am currently researching, but anyone is welcome to beat me to it) is the non-death of John Laurens in 1782 in otherwise worthless skirmish in S.C. If Laurens doesn't die there would be a Southern ally for Hamilton's programs making it much more palatable to the South.
In reference to the Duel, it really depends on how it goes down. I really don't think his surviving would make a difference, his next plan was according to Governeur Morris was a massive defense of Federalism explaining its true worth, etc. I could see him running for a Senate seat or Gov of New York, although then he has to contend with Dewitt Clinton and much more credible Burr (the duel is really what did him in). just some thoughts