When she asks why she can't live here, I say because u have to live with mummy right now until ur older. Is that the right thing to say?
I don't think it is your place to commit to things changing in the future, tbh - these are conversations that her Dad should be having with her really - it is lovely that she trusts you, but you can't influence or commit to changing her care arrangements, and in the long run, she may think that you have let her down, if things don't happen the way you have led her to believe. It may also fuel the hostility between the two homes if her mum finds out that you are talking to DSD about living arrangements.
I'm not a fan of allowing DC's to chose their living arrangements really - I always tell my DD that where she lives is a decision for me and her Dad to make, and while we will listen to her opinions, we will do what we think is best taking everything, including adult issues, into account.
It is an incredible hard situation, but I think that all you can do is encourage her to enjoy the time she spends with mum, and highlight the positives to her - no matter how grim it seems to you. She does seem to be managing to maintain a relationship with her mum, so perhaps there is an element of manipulation going on, but there is obviously a stark contrast between homes, and that is probably what she is struggling to deal with.
Can the school refer to a youth counselling scheme?