My thoughts, for whatever they are worth, are as follows:
1. AH's political career, at least as a potential candidate for elective office, was essentially dead by the time of the duel. His actions and wiritings during the election of 1800 had made him persona non grata with the entire Adams wing of the Federalist Party, his earlier admissions concerning the Maria Reynolds affair had marked him as somewhat unstable and unreliable on a personal basis and his actions as Washington's second in command of the New Army had alienated not only Adams but many others.
2. Killing Burr in the duel would have been seen by many as a bloody, retrograde type of activity. Many states, including NY had made dueling illegal (which is why the duel was conducted in NJ) so not only might AH have had some legal problems, much like Burr, but he would have been thought of as upholding an old form of "honor" which was rapidly passing away. I think AH would have easily survived the legal problems (as Burr did) but dueling was on the way out and the public perception of duelers was declining.
3. AH could have continued to be one of the leading lawyers in NY and the country and a powerful, behind the scenes, force in the Federalist Party. His mighty intellect and pen would have been at the service of the anti-Jeffersonians and I would dearly love to have seen what he would have said and wrote about the Embargo Act and other Jeffersonian-Madisonian half baked ideas.
4. I hope, but I can not say for sure, that AH would have backed younger, more moderate anti-Jeffersonians such as Dewitt Clinton and not stayed in lockstep behind High Federalists like Pickering. With AH's support, a broader, more populist, Federalist Party might have been possible and might have been a more powerful competitior to the Madison and Monroe Republicans. Perhaps younger men such as J.Q. Adams and Joseph Story might have even stayed Federalists if it was a viable opposition party.
Your humble servant
AH (with apologies to the real AH)