Maybe avoid the French invasion of Russia, and have the British devote more resources to the Peninsular War? That would leave even fewer British troops for the Americas and allow the US an opening to make some gains. It might also make the British more desperate for peace, with all resources being needed to combat the French in Spain.
Along with this, you could have the US fight a war with Spain in the early 1800s over Louisiana and Florida. This might give an incentive to keep the US regular army stronger relative to its OTL weakness, with the memory and lessons of a recent war.
Agreed, this would work. The UK still fighting Napoleon hard through 1816 means not only that the UK does not reinforce North America, but it may well pull out some troops. If combined with the USA negotiators being reasonable (not demanding all of Canada, allow Quebec to be free), it would get you an USA win. It is not so much the USA wins, as the UK is unwilling to fund enough regiments to get a draw.
As to the USA keeping a larger army, there would be some other ways, but all would involve military threats - larger slave revolts, more battles with Indians or the UK seeming more threatening.
If I was to write a TL, at first glance I would go with a UK diplomat/King making vague but threatening type speeches about the USA (think Kaiser Wilhelm II type stuff - bombastic and often meaningless) combined with Napoleon wintering at Vilnius instead of driving on Moscow. This should keep him in the war at least one more year. And with the American papers exaggerating the English statements, we could get a bigger militia. They still would be clearly class B or C units by European standards, but if big enough, we could get initial gains that are hard for the UK to reverse.
Agreed. Probably the only way the Americans could not only win the war but have it stay won — that is, not have to face Wellington and about 50,000 men a few years later — would be if the war against Napoleon ended in stalemate and the British had to stay focused on France. (Although it's hard to imagine Napoleon being satisfied with a stalemate.)
You can also get their by lack of will for the UK due to poor finances. The UK finances were quite stretched by the end IOTL, so just taking the war a few years longer may be enough to make a settlement likely. While Wellingtons army of 50K is impressive by American standards, it is too small to occupy the USA by quite a large margin. And the logistical/financial cost of supporting 50K men in the field is quite large. Since the UK chose to compromise IOTL and take a draw, they could easily accept the USA gaining some land.
For example, if early in the war, the USA gains Ontario and Quebec is either independent or also conquered by the USA, a treaty accepting the USA gaining Ontario and not gaining Quebec might look quite attractive to both sides. Or a situation where there are some "border clarifications" that "hide" the fact the UK loses lands. If the USA gains the north shore of Lake Superior, how big a deal is this really to the House of Commons?
The tough part is the USA gaining the early victories and holding the land for a few years.