But FOX probably owns the TV rights to the character
If FOX had the TV rights to the actual Batman, then they would've made an actual Batman show instead of a Jim Gordon prequel that tried to be as much like a Batman show as possible by tossing in every Bat-villain 15 years ahead of schedule. DC let them do a Batman-adjacent show, just like they're letting The CW do one with Batwoman and DC Universe with Titans, but the actual speaking, onscreen Batman is reserved for movies.
As I said before, my understanding is that it's not really about owning rights. It's not like the Marvel case where they licensed the rights to the X-Men and Spider-Man to different studios. There is only one studio, Warner Bros., that makes all the DC shows. They make them in partnership with the various networks and production companies, but the rights are all consolidated with DC/Warner, and DC/Warner doles out permissions for the use of its characters on a case-by-case basis.
And taht would make more sense as WB has allowed BOTH a feature film and TV version of both Superman and Flash - so 'Batman' wouldn't be an issue IF they had TV rights to the character and hadn't sold such to FOX for Gotham.
Again, that's not how it works. They decide on a case-by-case basis. I'm sure the only reason they allowed two coexisting versions of the Flash was because the TV show came first and was already a major hit by the time they began developing the movie character in earnest. If the movie character had come first, The CW never would've been cleared to do the show. As for Superman, he's pretty much only appeared on Supergirl at times when there wasn't a Superman movie on the way. Again, the fact that Supergirl is a successful show may also mean that DC/WB are more willing to give it leeway to use the character.