BFR is the what Musk called the whole system. We don't have individual names for the various spacecraft and the booster at this point.I don't think there was any ambiguity that the OP is referring to the BFR spacecraft visiting the ISS and not the booster.
I think each ITS ship should have a name like ITS Enterprise, or ITS Columbia. Tankers could have other names, like ITS Dallas or something like that. Cargo ships could have other names also. like ICS (Interplanetary Cargo Ship) Halsy. Since they will be reused, naming the ITS ships should be done. Boosters probably should be just numbered, like BFB (Big fricking booster) 001.
I don't quite know what was wrong with ITS as a name for the whole system and see no reason not to use BFR for the booster and BFS for the ship.
SpaceX names have all been based on birds of prey (Merlin, Kestrel, Falcon...) or dragons (Dragon, Draco). There are plenty of names available in those themes.They should just call it the Lammergeier and get on with it.
Ironic, all this effort to find the iconic name for something we hope to become the 747 of spaceflight
Any way, I think BFR/BFS will eventually be officially retired as a code name, and something more socially acceptable will be adopted...