Still, I don't think I can get past the basic character logic problem of her not agreeing with Batman's outlook, yet co-opting his image anyway. Seems hypocritical.
Not at all. You can be inspired by someone's ideas without feeling constrained to limit yourself to them. You can agree with their core elements but believe the originator hasn't taken them far enough, or isn't seeing the full possibilities. It's often that way with mentors and proteges, or with role models and those they inspire.
Besides, Kate's a soldier, and so I'd imagine that she fights with whatever weapons are useful to her. In Gotham, the iconography of the Bat carries power that's useful to a vigilante. So maybe it's a means to an end for her.
As for why Batman treats Batwoman differently than he treated Helena-as-Batgirl... I'm only partially conversant with the stories, but the impression I get is that Batwoman did more tangible good along the way, and Batman recognized the value of that. Perhaps Helena just didn't do enough good to earn another chance when she screwed up.