Blimey, I don't even know what I would do in this situation, the only certainty is your STBX is a fucking cunt.
I think the first thing I would do is sit with DSD and ask her why she wants to stay. The fact she would prefer to stay with you speaks volumes, it's either because:
A) she knows she will get an easy ride/closer to mates etc.
B) because she sees you as more of a parent than her dad.
Getting to the bottom of that would be crucial for my decision making. If she wants to stay for reason B then I would be considering having a leveller with her, setting some ground rules and going in on a trial basis with the understanding that she must leave if shit doesn't change.
If A, I would hold firm, you aren't a hotel.
You're in a hard spot OP, damned if you do, damned if you don't. I can't believe your ex is willing to walk away from his 16 year old daughter and start a new life, leaving her feeling second after his. Decisions made at this time could potentially impact her for the rest of her life. I know the responsibility falls on her dad but my dilemma would be:
"Do I dig my heels in because he's a cunt and she is his legal responsibility"
Or
"Do I attempt to carry on mothering this kid in the hope she changes if it's the right thing for her"
If I did agree to let her stay, I would be making it very clear it was subject to conditions and relative to change should she not uphold her end of the bargain.
If I decided she couldn't stay, I would be having that conversation with her myself rather than leaving it to her dad.
I would also (depending on age) see what the other kids think.
I just don't even know OP, I'm sorry this is happening to you and I'm heart sorry for this young girl too
.