This could get a bit silly, especially given how much Arrow has stolen from Batman over the years. But, Im interested in seeing how they handle it.
To be fair, whilst it's true that Arrow has taken a lot of Batman material, there's equally a lot of Batman material that they haven't used; Riddler, Penguin, Joker, Scarecrow.
Gotham also exists Supergirl's universe, so essentially there's 2 Gothams.
There is a version of both Gotham, and Bruce Wayne (and likely in turn, Batman) on at least two Earth's.
Ours, and Kara's.
Also, this is clearly a backdoor pilot for a Gotham-based show, right? Like, theyve probably been dying to do one but had to wait until Gotham was finished, and nows their window. Maybe Birds of Prey with Batwoman in place of Batgirl.
I'm not convinced that it is, at least not until Arrow is wrapped up. Having two vigilante shows run at the same time ... especially with the other CW Superhero shows already in the mix (along with Titans soon coming up). Might be a bit much.
If Arrow wraps at Season 7, then maybe, but I doubt we'll see a Gotham show on the CW at the same time until then.
Yeah, since they basically established that theres no Oliver Queen in Supergirls universe (or at least, shes not familiar with him) while there IS a Bruce Wayne/Batman, Ive pretty much developed a headcanon that the Green Arrow is Earth-1s Batman, while their Bruce Wayne is just an a-hole billionaire, and on Earth-38, Bruce Wayne became Batman while the whole Queen family died at sea many years ago or something, which is why Batman and Green Arrow share storylines/villains, lol. Setting this Gotham crossover on Earth-38 would totally satisfy my headcanon.
This actually makes a fair amount of sense.
Batman / Bruce Wayne is Earth 38's Masked Vigilante
Arrow / Oliver Queen is Earth 1's Masked Vigilante
Furthermore, this Batwoman doesn't actually need to be directly linked to Gotham and/or Batman if they (the writers) don't want it. We know that lore and background has, and can be changed for characters, so they may alter hers.
Edit: I've just read that she's a lesbian (at least, one of her incarnations is). I swear if they have a romantic scene between her and Sara Lance I'm going to be annoyed; that element of Sara and supporting characters has been seriously overplayed). Alex (Supergirl) likes girls too, but you don't see her jumping into bed with them all.