I think it it all depends on how long this war drags on.
As stated a status quo antebellum result is likely. But only if the war is short and the US acts smart in politics. Yes, the US is likely to take the Great Lakes easily and outbuild Britain there. But this victory can easily be undone if the war drags on and ultimately have very negative results for the US.
For starters there is an election within less than a year after the war starts. It is likely that we might not see any change and that the following Dem-Rep (probably Madison) nominee wins and keeps Jefferson's main policies going as IOTL. But midway through a war I can see a third term for Jefferson, which maybe not a bad idea at the moment but it might have negative repercussions further down the line.
Also even if the country is more united at first during the war, since there is no embargo, if the war drags on New England would still be loosing a lot of trade. Maybe even more so than with an embargo that was never quite enforced. So if the war drags on these states might find even more reason to rebel. There would likely be more support during the Hartford Convention (or its equivalent). If Jefferson's reelection did happen they will likely have even more reason to do so.
Now while the US might easily march into Canada, actually occupying it would take a much bigger force something the US might not be able to afford due to fighting elsewhere.
Furthermore as noted the British army is not stuck in Portugal and Spain at the moment, and while the US might dominate in the Great Lakes Britain is master of the Atlantic and the Caribbean. British troops will arrive in greater numbers. IOTL they burned Washington, with more men they can do more damage not just to the capital but major ports as well. New Orleans might suffer badly.
With more troops to deal everywhere, resources needed to occupy Canada, and New England on the verge of rebellion some Native leader (Tecumseh) will take the opportunity and make a nuisance of himself.
So the war will slowly turn a mess that would be very hard to clean up for the US. If New England does end up rebelling (not necessarily seceding but declaring neutrality until certain demands are met) you are pretty much killing the US very early in its life.
Now the interesting butterfly is with the British troops occupied across the Atlantic, what happens in Europe once they are needed in Portugal and Spain? Does Napoleon march victorious across Europe? And an America angry at Britain could easily ally itself with Napoleon.