Hamilton was already totally discredited as a politician by then, both because of the Reynolds scandal and because he had the tendency to pick fights with everyone who disagreed with him about anything. He's driven most of even his staunchest erstwhile allies away, some so far away that they literally went over to Jefferson's camp during the height of the Hamilton-Jefferson rivalry.
Also, Hamilton may have been the one challenged, but he was challenged because he was spreading some pretty insulting slander about Burr, and then haughtily rejected a sincere offer to publicly retract his remarks. If he shoots Burr, he will most certainly look like the villain. A complete jerk who slandered a guy, rejected all chances to make amends, leaving his victim wth only one recourse to keep his honour... and then shot him. (Keep in mind, Hamilton was a soldier, whereas Burr was not. What happened in OTL was likely a complete fluke, and had Hamilton indeed killed Burr, I think nobody would really believe it could hve gone the other way. It would be a case of a marksman shooting a man who rarely ever handled a gun.)
In any event, Hamilton's career was already dead, and after the duel, so is whatever's left of his reputation. I doubt he gets to be the subject of a totally a-historical, overly glorifying musical in this particular ATL.