The Sharples profile of Hamilton does seem to have been the best according to Hamilton’s family, but the marble bust by Giuseppe Ceracchiwas also a very good likeness. In fact, when John Trumbull was commissioned to paint a posthumous portrait of Hamilton, Eliza lent him the bust to use as a reference for Hamilton’s face. That 1806 John Trumbull portrait went on to be the basis for the $10 bill.
How Hamilton ended up with the bust is actually kind of a funny story–Ceracchi was a sculptor from Rome who had proposed to Congress a memorial to the American Revolution. Congress ultimately rejected his proposal, but not before he’d sculpted several prominent figures, including Washington, Jefferson, and, of course, Hamilton. Ceracchi sent the completed busts to each of his models once it was clear the memorial wouldn’t happen; more problematically, he also sent them each a bill. Where Washington determined to return the bust rather than pay, Hamilton recorded in his cash book $620 “for this sum through delicacy paid upon cherachi’s draft for making my bust on his own importunity & as a favour to him.”(See Ceracchi to Hamilton, 16 July 1792, fn.2)