It's 'the' batcave. Bryan Q Miller made that very clear in the last two issues of Batgirl itself.
I don't think that it's going to be Bruce who's upset about the women in the batcave. And that's something that I really dislike about it--the women versus men fake feminism focus that I believe may be upcoming. Hopefully, I'm wrong about that, and that it won't be a women against the men deal. For Dick's own home.