I'm coming at this from the mum of three adult kids who is also a teacher of mainly year 11 students.
Firstly, I would have taken a bullet for each of my kids in a heartbeat but there were times in their mid teens when I didn't like them. Doesn't matter, because in those rough phases I still loved them unconditionally.
Secondly, the top set year 11s that I teach are a walk in the park. Lapping up their Shakespeare, completing extension tasks, on course for straight A's (or the new equivalent) next August. Great. The lower sets are a challenge, many struggle, are disengaged, have behaviour issues, various barriers to learning. They will, if I engage them, inspire them, and teach them, walk out of school next summer scraping a C grade. Those are the kids that make my job worthwhile. Those are the ones who I'm most proud of.
It's easy to teach bright kids. It's easy to parent 'easy' kids.
As a parent you do not have the luxury of choosing to parent!
She's your SD, she is entitled to live in her father's home. You are entitled to be exasperated with her, but every day she should feel loved and valued by both of you. You seem to be taking on the lion's share of the parenting issues here. Why? Does her dad see things differently? Or is he not as involved as he should be?
It is glaringly obvious to anyone reading, that this girl is dealing with rejection from her own mum, that anger is fueling her reticence to reconnect with her mum. She will of course, along with all the usual teen angst, be feeling hurt and unwanted. She is very probably jealous of her step sister, because she lives with her mum.
She needs encouragement to seek help with her MH issues. Your early comment about her not going to an appt. was 'make of that what you will' which sounds extremely dismissive. What I make of it is that she doesn't feel worth help. This needs to change.
Her father either needs to step up, or you need to back off.
The worst thing you can do is try to instigate her moving back to her mum's, because she will feel, yet again, unwanted and unloved.
Please stop assuming that your little girl will turn into a dream teen because of the way you are bringing her up. It is at best naive, and at worst offensive to all the parents who stick by their kids, day in day out, who feel they are to blame. Sometimes it really is about where the dice falls.
Finally, please remember love is a verb. You don't need money to love her.