Battle actually occurred after the treaty to end the war was signed, so ending would have been the same. Comms were just slower in the day.
The Brits would have kept the port of New Orleans. There is no chance they would have given it back if they had won the Battle.
This would have effectively removed one of America’s greatest strengths, our incomparable river system to the great farmlands and what became the great industrial centers.
George Friedman has written extensively about this.
If you think the British would have returned New Orleans based on that technicality I have to disagree.