BBC actually alternates between Derry and Londonderry. It has done for some years now - since 1998 or so, I think.
People also don't have to be hardcore anything to use one over the other either. It's usually outsiders who perceive it this way which doesn't necessarily reflect the local population itself. Especially as you'd likely perceive the Apprentice Boys as hardcore Unionists but they have "Derry" in their name.
When I have occasion to mention it, I'll say Londonderry since that IS the official place-name. It's not often I need to specifically say the name though - other terms can do, e.g. Foyleside is another you can say or a local might say a district name such as Waterside. I might alternate between the two for a Nationalist but not for a Republican.
Basically, it's context specific and dependent on the audience.
The music for "Danny Boy", for example, is actually called "Londonderry Air"! The lyrics written by an English man is the bit that is actually called "Danny Boy".
That's why you'll see the title as "Londonderry Air (Danny Boy)" or other way round.
I don't recall any major disagreement over that either.
The chamber of commerce there is actually called Londonderry Chamber of Commerce.
There is the Derry Journal whose political alignment is Nationalist and the Londonderry Sentinel whose political alignment is Unionist.
You can find examples of both in various places including business names. So, it's not really about hardcore this or hardcore that.
It's dependent on various factors and another might be someone's mood on a given day which a joke was built around where the punchline is "it depends on whoever I'm talking to", i.e., mood and audience informs which name to use.
A visitor could do what a lot of people in Northern Ireland do when they want to avoid saying a place-name.
Just call it "this place" or "that place"! 😂
Walled City is another name.