I always thought he was lying with his dieing claim to have thrown his shot away. And, if the claim were true, he would have known Burr would kill him.
Signifying to the end, adding to the character assassination started by Jefferson.
Burr was scum but Hamilton was not much better. Back when we were kids there was a close look at the weapons used in the Duel.
I have always thought he went for the Main Chance and grabbed the Gold Ring. Did some good things but layed the ground work for future destruction of the Republic.