There is no record of what happened to the bullet in Hamilton. I addressed this here. The trail goes cold with the autopsy report.

As for the bullet Hamilton shot, that was never recovered. It did not lodge in the tree, but Pendleton, Hamilton's second, claimed to have returned to the spot and:

Mr. P having so strong a conviction that if General Hamilton had fired first, it could not have escaped his attention (all his anxiety being alive for the effect of the first fire, and having no reason to believe the friend of Col. Burr was not sincere in the contrary opinion) he determined to go to the spot where the affair took place, to see if he could not discover some traces of the course of the ball from Gen. Hamilton’s pistol. He took a friend with him the day after General Hamilton died, and after some examination they fortunately found what they were in search of. They ascertained that the ball passed through the limb of a cedar tree, at an elevation of about twelve feet and a half, perpendicularly from the ground, between thirteen and fourteen feet from the mark on which General Hamilton stood, and about four feet wide of the direct line between him and Col. Burr, on the right side; he having fallen on the left. The part of the limb through which the ball passed was cut I off and brought to this city, and is now in Mr. Church’s possession.

The bullet, according to Pendleton which is the only attempt of an accounting we have of its path, cut through a branch and continued on its way to who knows where. It was never recovered either.