This is competely what if question. Let's say that WB did alone them to have bat Girl to appear on SUper Girl season 4 how would you introduce her

It'll never happen.

But if it did, I'd say you introduce her as Barbara Gordon, in town under a false premise, when really, she's investigating something. Maybe she's a guest character for a few episodes, but don't reveal her as Batgirl until episode 2 or 3.

Given that we've already got half a dozen or so heros who have gone "into the field" with some kind of "meta" abilities (alien physiology based or otherwise), it would probably be more workable to have, for example, Winn trying to get information on a case online and suddenly an avatar shows up on his monitor.

"Like to help. You can call me Oracle".

Smiley Face.

Or, given some comic history of how Barbara "borrowed" satellite capabilities from Lex's space systems, Lena could be trying to trace a leak at L-Corp and come across a Gotham City connection leading back to a certain clock tower.

If they wanted to follow up, later on they could have Winn or Kara "coincidentally" bump into a red-head in a wheelchair, who makes remarks that steers them to the bad guys.

This could let the show technically follow whatever "ban" may be in place because of Fox's Gotham apparent exclusive permission to use blatant named references to Bat-characters, and still give fans a fun episode.

I think Oracle fits the current CW better than the whichever "restored" Batgirl version they might use (even though I am eagerly anticipating a butt-kicking, line-swinging Barbara Gordon that Joss Weadon is going to be presenting in his movie version).

By first stating Batman exists.

I liked her in the crossover with Supergirl comics searching for Psi, I would much rather her be more near Kara's age.