I've talked about references to the Batcave on other TV shows before. (This link to the latest reference also includes a link to the first article.) And its notoriety has an international following.....
When George Sands told Annie that he was treating his severe case of "moping" by hunting vampires, the ghost argued: "I meant 'take a shower', not storm the Batcave!" (This occurred in the fourth season opener of the original 'Being Human'.)
It's not something I look forward to, but should the day come when Adam West passes away and some network decides to bring 'Batman' back as a live-action series, I think a splainin could be drafted in which both versions of the show exist in Earth Prime-Time. But so long as the actor lives and breathes, the original Bruce Wayne does as well. And that means any new version of the show must be relegated to an alternate TV dimension.
To consider any other option would be... diabolical.