Okay, I really don't like Sophie. She would've come clean with her full romantic history if she were over Kate, but she hasn't and she's not.
Mary on the other hand is growing on me. She's funny and kind of ditzy but still smart.
I know it's only season one and very early, but I am having similar problems with Batwoman as I did with Supergirl. WAY too much mention of the male hero counterpart. Hush is a very personal villain for Batman because he is Bruce's childhood friend and knows about his dual identity. So using him in this episode just means the presence of Bruce is hanging over them. Kate needs her own rogues gallery (we know a handful of Bruce's villains are in Arkham, such as Riddler, so keep them there). Bruce does need to make some kind of appearance at some point. I wonder if Kate will get a moment like Kara did where she managed to beat Superman who thought she was Zod. That seemed to help Kara distance herself from Superman.
Also, I guess Gotham or just Vesper never heard of Batgirl. Did she suggest "Batchick"? That's not very PC. What about Batperson?
Self destruct command seems more like his style. I remember him doing it with his utility belt at some point.
I kind of want to go back and look for all the previous Batman references. The most blatant I remember is Harley Quinn being on the Suicide Squad.
Bring back Huntress! Oh yeah, there's another Batverse character who was on Arrow.