
There have been rumors of Batwoman on Supergirl for a while now and Supergirl has already crossed over with Flash and Green Arrow, although they established that Supergirl is from another Earth seperate from Flash/GreenArrow. They also turned Supergirls sister gay very quickly and she is a prominent character on the show, and also is kind of a bad-ass cop. I don't know? Superman appeared in the first two episodes of the Season2 premiere and never showed up again, so I don't know that DC would allow the use of her (Batwoman) characters, or others from Gotham unless they take creative license with it. You can read it here:
https://moviepilot.com/p/supergirl-is-alex-set-to-become-batwoman/4142877
Unless you meant Batgirl/Barbara Gordon, who is part of Birds of Prey, in which case would fit better into Flash/Arrowverse, but I haven't heard much about that rumor.
As for Archie, I've read ppl comment that they made Archie and Jughead really horny teens?