OP, she might have been very chatty with you on the handful of occasions you've spoken to her, but I could be very chatty with acquaintances, too. It doesn't mean I want them moving into my personal space.
She mentioned the separation, but I wonder if a large part of the problem is actually due to a lack of familiarity. You've been in a relationship with her dad for 3.5 years, not her. You're effectively just some random woman who is apparently going to share her bathroom, her kitchen, her living room, etc. Those are quite intimate things - no one likes sharing with strangers, but they put up with it for close family (like mothers), but you're not close family.
I think you should drop the overnights issue and try to have some short but quality time with her on the weeks she is there. For example, you could go out for lunch with her and her dad and then disappear so you get some interactive time as a group, but they get the rest of the weekend together, and her dad/daughter time is preserved. Let her get used to you more before you broach entering her safe home space again.
Even when she is old enough to move out, her dad's home should still feel like her home, so I think it's worth taking it super slowly so she doesn't feel pushed out. It would be awful if she left as soon as she could and refused to stay over at her dad's because she didn't feel comfortable there.