Batman is Important For Batwoman's Origins, But Not Essential
The simplest solution would be for Batwoman to be based on Earth 38, with Warner Bros. making no objection to The Arrowverse using Batman this one time for the purposes of establishing the character of Kate Kane. Unfortunately, what is simple is not always easy, particularly when legalities are involved. This begs the question of how vital it is that Batman appears to establish the character of Batwoman at all.
Despite Kathy Kane being a cousin of Bruce Wayne and her name and costume clearly being modeled on Batman, there is surprisingly little connection between the two characters in the comics. The two are far from close kin, with Bruce largely avoiding extended family gatherings. Bruce only makes token appearances on the social scene in Gotham City, whereas Kate was enough of a party girl and press darling in her youth to make it difficult for her girlfriend Rene Montoya to avoid the limelight and keep her own status as a lesbian secret from her family and her supervisors at the Gotham City Police Department.
The two also approached vigilantism from different angles, with Kate's training as a fighter coming from her desire to follow in her father's footsteps as a career soldier. This dream was denied to her after her relationship with another female cadet was revealed while she was enrolled at the United States Military Academy at West Point and Kate resigned rather than violate the honor code by lying about her sexual orientation. Kate was later inspired to vigilantism after a brief encounter with Batman, where The Dark Knight followed the sounds of a struggle to find an undistressed damsel, following Kate's using her combat training to turn the tables on an attempted mugger.
Kate began fighting crime on her own after that, using military gear taken from her father. When Colonel Jacob Kane discovered what she was doing, he stunned Kate not only by voicing his approval but by arranging for her to train with some of his old army buddies. When Kate came home to Gotham City, she found that her father had commissioned an armored Batwoman costume for her, along with a host of other special tactical weapons combining the finest in military hardware and prototypes based on gear it was believed Batman used. The two would not learn that Bruce Wayne was Batman until long after Kate had taken up the Batwoman mantle.
Can Batwoman Exist in Gotham Without Explaining Batman/Bruce Wayne?
So the only requirement of Batman's existence in relation to Kate Kane's becoming a superhero is an acknowledgment that Batman exists. Batwoman's costume and name were entirely derived from Batman, but beyond that the presence of Batman and Bruce Wayne is inessential. It would be simple enough to acknowledge Batman's existence with a quick description of Kate Kate's inspirational encounter with The Dark Knight before moving on and never mentioning him again.
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However, it may be possible to try and introduce Batwoman into a reality without Batman, painting Kate as the original hero in the reality of The Arrowverse. One method of accomplishing this might be altering Kate's background so that her defining moment in deciding to be a vigilante is the violent deaths of her mother and twin sister during a hostage crisis when she was twelve, rather than a chance encounter with Batman after an attempted mugging. Unfortunately (and ironically), such a bid to make Kate stand apart from her famous cousin would only serve to make her origin seem more derivative of Batman and eliminate much of what makes Kate Kane unique as a character.
One fan theory has suggested that Bruce Wayne may be paralyzed from his encounter with the villain Bane in the Arrowverse. If that's the case, then Batwoman could be presented as a legacy character, taking up her cousin's battle in his name and image. It's a unique idea and would be an interesting twist on the classic story Knightfall. Unfortunately, we again run into the problem that such a story would only serve to make Kate Kane into less of a unique character and everything Batwoman's producers have said so far suggests that they will be staying true to the spirit of the comics where Kate is very much her own hero.
The simple solution of mentioning Batman and how he inspired Batwoman without showing him seems to be the best option. It would also fit with previous statements by The CW president Mark Pedowitz, who confirmed that Batman does exist in the Arrowverse but that there are no plans for him to make an appearance in any capacity.; that does not outright eliminate the possibility of dropping his name in the same fashion as earlier episodes.