I don't think it would be very satisfying to have Kate get fridged off screen. I think they implied that Batman finished off the Joker in Batwoman so that would be messy too. It would be kind of interesting if Harley came to finish the deed on Kate for revenge though that sounds like a potential later season had Kate stuck around.
EDIT: It occurs to me that it wouldn't have to be off-screen necessarily if we see it happen to the character in costume. It would admittedly allow for a transition of the character though I'm still not sure about that approach.
I think that would be interesting for season 2 with introducing Harley Quinn. They probably can't due to the DC films version.
If it could be worked out then Harley kills Kate before season 2 starts and Gotham is worse after Batwoman gets killed.