He'd try - just as he did in 1805. He'd probably make a pigs ear of it again, given that despite being an islander himself he really didn't understand that ships, at the mercy of wind and tide, can't be ordered about to precise timetables the way armies can. The result of this, combined with the inherent superiority of the RN (constantly at sea, and therefore constantly worked-up) over the French and allies (stuck in port, and non too experienced to start with), would almost certainly have been another defeat in detail.
I think you are underestimating what the Continent can build. Without the interruption of Continental troubles, the French and satellites would have over 300 ships of the line. That's a lot of naval power. Or, if you suck at sailing (since no one but the Danes and Spanish in Napoleon's sphere of influences have many left), glorified floating artillery that gets destroyed by the enemy's first barrage because they can't get out of the way of their enemy's guns. But that's a lot of glorified artillery. Can the RN pull off another victory? Well, possibly, and if Napoleon actually takes personal command probably. But it will be harder than the last time.
Control of territory simply wasn't relevant.
I disagree. America can grow food, and there is stuff for powder. So if they didn't mind being 100% broke, they could go into Canada IF the British needed to cannibalize manpower from Canada (as we already noted, the Canadian-British were doing quite well as it was). Britain cannibalizing manpower from Canada could result in Western Canada going to be occupied, while the entire American nation gets struck behind a blockade and falling to a 15th century standard of living. You really only need powder, shot, and food to continue a war at this time, the rest of the economy doing to the dump doesn't matter if the nation is willing to put up with crap living just for the war. Which America did put up with in the 1770s (minus the Loyalists and neutrals who of course didn't like crap living conditions in Patriot controlled territory).
My point is that America is unlikely to gain territory, but territory is important. if they got it.
It is for this reason that I think the many people who think the worst case scenario for the US would have been an early defeat of France have got it the wrong way round. On the contrary, an extended European war would have delayed the onset of the asymmetrical insurance rates that led to pressure to end the American war, potentially long enough to induce the unconditional surrender of the US due to the collapse of its economy.
An extended European war will mean the British central government isn't putting much effort into taking American territory, which I mentioned is important too. With a European war going on, the British aren't likely to do more than take Ohio, some Western territories, and call it a day to let the RN put some economic pressure. There would be no further land operations, if the British got that favorable result they wouldn''t push it. And if they got unfavorable results, they obviously couldn't push it.
The troops would be in Europe in Iberia, or perhaps waiting to counterattack a possible invasion. A protracted European war might not lead to an America victory, but it certainly wouldn't lead to a British victory. Any time British saw they were winning in the American theater, they would probably accept a "ok, how about you take our Western territories and some reparations?" Far from unconditional surrender, the British would be happy with this if Napoleon is causing trouble in Europe.
This is the "European War is going on and Britain is fine" scenario. A "European War is going on and Napoleon is looking scarier" would be better for the Americas. They key to an American victroy is making Parliament think it needs to tap into Canadian manpower for the European front. If we get that, a grift can form. Since War of the Third Coalition PODs like Trafalgar are out (based on the timeline we are working with since impressment was the casuis belli), the best way to make Napoleon scary is a Russian campaign success. That alone probably isn't enough for them to resort to conscription, but if that doesn't make Napoleon "scary enough" I bet a successful Russian Campaign and that giant fleet that was under construction all over the Mediterranean (30 ports! And that's only counting where the ships of the line were being made) was like "really scary and juuust slightly less scary than imminent invasion"