who gives a crap...the war was over..
No, the treaty was signed, that’s all. It took months for anything to travel over the ocean and NEITHER New Orleans nor British forces in the Gulf new anything about a treaty.
So even if Britain consented and the treaty could zoom over in 2 weeks, it was not ratified by the other country, either. It still would not be officially over.
Even if the combatants had some inkling of this, the battle may have made a difference. It may have made the Brits angrier and tear it all up (which still would take much time to make clear). Or make them give up altogether.