Canadian nationalist sentiments aside, Canada's fate here is entirely dependent on how Britain fares. If by some miracle Napoleon manages to somehow stabilize the continental system long-term, lock down control of Europe, and then cut off and bankrupt England...
OK, this requires further hypothetical. Boney offers the King of Prussia Emperorship of Germany (including those rich Austrian lands!) for alliance, and through skill and battlefield luck, they manage to wrap up Germany and get western and Central Europe under joint control fairly early. Somehow they convince Russia to join the band in exchange for, I dunno, political stake in Hungary or something? Enough that Europe is reliably under Boney's control. So he can focus on bankrupting and invading Britain.
So America says "yo Britain stop impressing our dudes", Britain says "fuck you, we need dudes", America says, "don't care, those are OUR dudes, asswipe", war breaks out, but then Boney invades Britain and it turns into a clusterfuck.
America slowly grinds up into Canada (at the time, America was only somewhat more powerful than Canada, not overwhelmingly so, so the Ontario peninsula is going to take some major work. Napoleon offers an alliance to the Americans complete with lucrative trade deals AND some nice new land out West at fire-sale prices! America has money, wants trade, doesn't mind some cheap land, Boney needs money, is trying to build a trade bloc to cut the Brits out of the loop, doesn't really need a big American empire that nobody really cares about anyway and is populated mostly by Spaniards and lots and lots of natives. Match made in heaven.
The Feds finally get it through the states' thick skulls that an organized military is an actual thing that the USA NEEDS. American national identity starts to form, much like OTL.
Southern states are really not on board with taking the Ontario peninsula and probably the northwestern shore regions (Nova Scotia, Newfoundland, etc.) because that cuts into their unfair Congressional advantage. But with the French quite likely offering a sweetheart deal for the English-speaking areas in exchange for collaboration on the lucrative fur trade and France keeping Quebec, well, money talks, and the South can be mollified with Louisiana and slave-power control over the Mississippi delta.
Expansionism is likely more of a thing and goes further--more of Mexico is taken, probably bits of Central America and the Caribbean, too. The North is more powerful earlier and probably industrializes a bit quicker due to higher population and a somewhat stronger economy.
We end up with an expansionist, French-aligned, abolitionist for reasons of Northern economic practicality USA.
I'd prefer OTL's "war was a draw, won the peace".