Burr's intentions are unknowable. He kept no paper records and confided very little in very few people. In fact, we know him chiefly through Madison's diary and a few court transcripts.
Hamilton and Jefferson had already destroyed Burr. His actions in the short term are unlikely to be any different from OTL. And it's unlikely that he feels any NEED to kill Hamilton, now that honor has been satisfied and the duel conducted. Hamilton will relent after 1 shot. Burr will come back on stage only years later, as an oldman, presuming he makes up with various people, Hamiltonincluded, which is not impossible.
The more interesting and important half of the question...whither Hamilton? His own papers suggest that he actually regarded Burr as in the right...he had destroyed a patriot, statesman and countryman with slander, and the challenge from Burr made him confront the consequences of immoderate and untrue speech. The Federalist Party, though,still regards Hamilton as its saint and leader. Where will he lead it?