One of the great things in our opinion about the Arrowverse, is that it gives the ability for non-trinity characters (Superman, Batman, and Wonder Woman) to shine. That said, there are some characters that have deep ties to their more popular counterparts. For example, Superman and Supergirl. Supergirl needed to address the Big Blue Boyscout who has such a powerful connection to her orgin as Supergirl. That doesn't seem to be the case for Batwoman according to Entertainment Weekly. The CW boss, Mark Pedowitz was quoted saying this:
There’s no discussion about a series. Batman already exists in the Arrowverse because last season, Oliver Queen mentioned his name at one point. And Batwoman, if the series goes forward, lives in Gotham. There’s no plan at this time to have Batman appear.
Now this answer was in reference to if a Batman show was in the works, but his answer may give us much more than that. Because of the link to Oliver Queen, Batwoman is likely on Earth One. Additionally, with no need to include Batman in the Batwoman show, our discovered test dialogue may indicate that Batwoman is replacing Batman, at least for the time-being!
Let us know what you think in the comments below and in the forum!