If Alexander Hamilton leads an army into Louisiana and Florida, then I think you'd see a second term for President John Adams. The Federalists' stock goes up in the western states, with Hamilton as a conquering hero. I think an enunciation of principles, especially internal improvements like roads and canals, would be a logical way forward. The war and the internal improvements platform captures the political loyalty of young men like Henry Clay, who will go on in OTL to found the Whig Party and in this TL will become a Federalist.