This sounds like two separate PODs, really. The latter one (Laurens surviving) is of much greater interest. Hamilton, although obviously a very talented man, had a way of making enemies. Even without his scandal, his political career was probably going down the drain unless he found a way to stop pissing people off (which would be unlike him). President Hamilton is not going to happen in most realistic TLs.
Laurens, on the other hand, was by all accounts a personable and well-liked man. Also, a southerner, who was also politically very much on Hamilton's page... and anti-slavery. He could be the kind of man who could turn the Federalists into more than a (mosly) northeastern party. An ideal presidential candidate if you want a USA that goes really Federalist and acts against slavery (or tries to) at an early stage.
There was a fun TL (although I'd say it was on the rosy side, being a bit optimistic about how much Laurens and Hamilton could realisticially achieve and how politically inept the enemies would be) about this exact subject years ago. It was called
Laurens Lives. Mind you, this TL was discontinued when it got to 1808. it hasn't been updated since 2011, and the author hasn't been only sice 2012. So it's very much dead. But it's exactly what you're looking for, and the discussion in the thread is great to read. (I ony rwad the whole thing after becoming a member here, which was after it was already discontinued, and it was very much worth the effort.)