The same way it left loyal British citizens to be occupied permanently after the Revolutionary War - the loyal British citizens either stayed or left the country.
Still, the US winning will take a PoD going back at least a decade - increasing the scope of the Army to actually exist in reasonable numbers prior to the war. probably would take a Federalist leadership (though that same leadership would have not lead to the War anyway considering their pro-British views as it stood). The only other option would be another conflict - perhaps a border one against the Spanish over the Louisiana territory - that, or a war against France to take the territory (the Quasi-War expands beyond its extent OTL, combined with the superficial appearance of success in Haiti (there doesn't have to be real success, but even the chance that France remains in North America)), could push the US to attempt war against France, which is a war that they'd be far more insulated throughout compared to a war against the UK. Either way, it'd provide a core of younger recruits with more experience that could be called back to service and another increase in funding that would create more reserves for the US (heck, if they take the Rio Grande claims, as difficult as that might be, they'd have even larger borders than OTL which need defending. They especially have to worry about the Comanche et al, so more Indian Wars will translate to more Army funding, etc).
But that being said, with the forces on the ground, with a change at the start of the War of 1812, I see no way to majorly affect the end results, which is status quo ante bellum for the US - they simply have too little to work with at the start of the conflict.
The easiest border change that I don't think I've seen mentioned are earlier finalization to the Minnesota/Maine borders. Though, again, I don't think that could be achieved OTL, but in some sort of conflict where the US starts out with a reasonably-sized military as mentioned above.