It's a silly shibboleth, but I automatically ignore anyone who calls it the Department of War or Gulf of America. Hasn't steered me wrong yet. They're telling me they're the kind of people who only care about defending fascism.
I think it's worth giving people a tiny bit of grace on this. I've surprised people by explaining that the "Department of War" is just fascist fanfic and that the legal name has not changed.
It's a testament to the broken information ecosystem we're in that many people genuinely don't know this. Most will correct themselves when told. I agree with you that those who don't are not worth engaging.
Google Maps calls it Gulf of America, pretty difficult to ignore Google.
I call it department of war, because I think it is a great self-own on their part to do such a rename.