The British had a difficult time winning on the offensive
against the US in prepared positions in 1814, as the results of Hampstead Hill, Plattsburgh, and New Orleans all make clear - so, how exactly, does Pakenham do any better than he did historically?
Much less better than Ross and Prevost did, in their theaters?
Don't forget, Wellington had already advised Liverpool to make peace before the news of New Orleans, when the Duke said "thanks, but no thanks" to the American mission that Liverpool had offered...so the British had already made their decision for peace before the news of New Orleans.
Two decades straight of fighting had been enough, thank you very much.
Best,