The most plausible POD is probably concerning the Chesapeake Affair - 1807. IOTL it led directly to the embargo acts, and public outcry in America greatly exceeded in 1812 - even New England was for war.
The reason(s) this did not lead to a war ware highly debatable, including Jefferson being President. A nice POD, in my opinion, is to alter the Burr affair slightly, have him fail earlier or not attempt to steal the southwest. As it was his trial was going on during the Chesapeake Affair, and Jefferson wanted only one problem at a time.
Idea!!! Have a nice early POD butterfly away the Hamilton-Burr duel. Burr isn't tainted, so stays Jefferson's Vice President and doesn't instigate intrigue in the southwest. Hamilton remains a great leader of opposition for the Federalist Party. If the war goes badly enough he can do....just about anything frankly (criticism of fiscal policy?) Another direct butterfly is that Andrew Jackson does not associate with Burr, therefore his patriotism is unquestioned - therefore he gets a General's commision almost immediately when the war starts, rather than being shelved for a year+.