I like your idea.
Sometimes you can keep someone safer by sending them away. Sometimes you keep someone safer by keeping them close.
Bette really does not have time to wait for development in the pages of Batwoman. We've been waiting for decades for DC to remember Bette exists. She really needs to bloom now...so other writers may become interested in her.I would like for Bette to be tied into other things than just the current Batwoman. Some of her possible true but not reconciled history involves time with the Teen Titans.
In effect Bette/y was trained by Batwoman and maybe to a small degree by Bruce, in the Silver Age.
Possibly some training or just working on her own skills in her brief Teen Titans stint (or similar retconning)
Then her best training would seem to have come in the new 52 era with first Kate, and then Jacob.
So on the idea that she may have more skills or training than most would think; ie; "appears to be *just* a blonde" I could see her confidence growing and allowing herself to be the 'plant' for certain missions.