If Quebec was taken in the War of 1812, does Britain surrender BNA, or send the Duke of Wellington?

With the exception of Isaac Brock, practically all of the British officers stationed in Upper and Lower Canada wanted to fight a defensive war against the Americans, rather than push into American territory during the War of 1812. Governor General Provost was willing to surrender much of the western territories and Upper Canada if Lower Canada could be held on to, as Quebec served as the main entry point of British reinforcements and material aid for the war effort. It ultimately didn't come to that, as early victories in the west at Michilimackinac and Detroit sustained the western front for most of the war, and a win at the Battle of Queenston Heights prevented a loss of the middle (at the cost of Brock's life).

But, let's say that, through some means, the British/Canadians lose Quebec by 1814. Perhaps Brock is appointed to the Peninsular Campaign instead, and Provost and General Procter agree to retreat to York (Toronto), where they are further pushed back and Quebec and Montreal are eventually taken. At that point, do the British consider most of British North America a lost cause, ceding it the the Americans with the exception of the Maritimes, or do they launch a full offensive led by the Duke of Wellington? IOTL, Wellington said he was willing to go to Canada in late 1814, but considering it was more or less a stalemate at the time, he felt that a status quo antebellum should've been pursued, and that his skills would be better put to use in Europe. His attitude would naturally be different if the British were distinctly losing, but I don't know what course of action he would support.

If the British do fully supply a counter-attack, do they just attempt to reclaim the Canadas, or figure that while they're out there, they should push their claims into American territory, such as in northern Maine or west of the Ohio Valley? If Wellington is campaigning in Canada, how would that affect Napoleon's return and the Hundred Days?
 
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