It might not be ASB, but it's still pretty far out there. Hamilton wasn't just unpopular nationally, but within his own party as well, having undermined Adams' re-election bid in 1800, among other things. And the Federalist Party was on a precipitous downward trend after that year, so the odds against him beating Madison seem astronomical. And even if he pulled that off, he'd need New England's support to do it, and that region most strongly opposed the War of 1812, so he would be foolish to pursue war with Britain. And again, even if he'd gone to war anyways, the President wasn't a terribly powerful position at the time, so there's not much he could do to improve the military in just a few years. Conquering Canada is right out.