There's nothing to discuss. It's the Gulf of Mexico, and suggestions to the contrary are to be appropriately dismissed as childish.
Aa a side note, make sure that, no matter what, you find a way to follow "Gulf of" with "Mexico". It should flow by rote in the hearer's mind, simply by virtue of the fact that they hear it the correct way when you talk about it.
This is a trivially easy battle to fight. Only use the correct name, and it will be the one that sticks. Using the incorrect name, even to make correct statements about how it's dumb and childish and not going to be its name going forward, reinforces the lie.
Mexico is the Gulf-of-havin'est thing you ever did see.