How do you reckon that? It is well known that the treaty of Ghent was signed before the Battle of New Orleans, and you are saying they would have said ‘Oh, on second thoughts....’
Doubt it.
In any case, a few days after the Battle of New Orleans the same British force (under more effective new leadership) successfully captured Fort Bowyer and were within an ace of capturing Mobile, Alabama when news of the treaty arrived, whereupon they immediately vacated the field and the fort, the same would have happened had they received the message had they successfully captured New Orleans.
IMHO the spectacularly one-sided nature of the Battle of New Orleans has been seized and contorted by biased historians as being more significant than it actually was, and is used as a distraction from the fact that the US did not achieve it initial aim of conquering Canada and ended up having to spend the rest of the war on the defence trying to stop the British rampaging through the US...
But it did give us a great song by Johnny Horton. That alone makes it worthy.
The Treaty of Ghent had been signed but not ratified. The treaty could have been discarded if military events had been different.