OP, I've no idea if any of the following is at all useful but FWIW...
It's quite early days, as a child a friend lived with us for 3 months before returning to her family. She may not have time to properly miss you yet. It could all feel new and exciting and she is enjoying this new found independence. So for this reason I wonder if you should pretend that this situation is permanent to encourage her to face some hard truths about this choice she has made. So for example, asking if the mother at the house intends to pay for all extra food, water, clothing sanitary products or if your DD intends to get a job after school and at weekends so she can still go to school (a legal requirement). I think you still need parental consent for a child to work if under 16 but you could check on the laws on that. Does she get an allowance direct to a bank account, so you pay for her mobile phone? There are consequences to being independent.
You could also suggest she comes to get some more clothes and move more personal belongings over there if she is moving out. She will have to do her own washing and sort her own school uniform.
Are there any siblings she will be missing or are they missing her? Any pets? I would talk to her about the status quo at home and remind her about the family life/routine she is missing out on, roast dinners, walking the dog etc...anything to remind her of home comforts and mildly suggest that life continues, doesn't she want to be part of it? That sort of thing. It also might give you a chance to talk about things not related to trans or her being away and be a way to encourage a different type of conversation which can be 'friendly'.
With the mother I would explain clearly that you have parental rights as well as responsibilities and that you have informed the police. She is obviously not in a kidnapper and I am not implying that but you can remind her that you are the parent and if she is to wants to take on any rights and responsibilities she needs to apply for a type of guardianship. This makes it sound very serious and I suspect that her "I don't want to get involved" stance is going to be increasingly difficult to hold.
The person your DD is dating is another part of this and I don't have any great ideas there. I'm not sure it's not best to ignore that and try to maintain and extend your bond with your DD in other ways. Invite her for a coffee or to go shopping? Generally I think parents shouldn't be their child's friend as they are the parent but if a hard line of "come home now or properly move out" isn't going to have the desired result then maybe gently stepping back whilst consistently saying you want her to come home might work.
I say all this with the greatest sympathy, I'm no expert and can only hope this is short lived and she comes to her senses.