Don't lose hope, hang on in there! She needs you. She needs you to give her the boundaries that she's either lacking the confidence to assert herself or that she's too indoctrinated to see for herself.
Keep away from trans chat as much as possible, like another poster said, do the whole, "that's nice dear, did you know Uncle Derick has a new kitten?". Show her there's a whole life out there to be explored, she doesn't need to have her entire sense of self and identity wrapped up in this relationship and trans issues. Send her pictures and info of exciting/homely things that you're doing without any pressure to come home (but obviously you can let her know she's very welcome). If there's no trans talk and no pressure, there's nothing to kick back against.
The other mum is clearly facilitating this. She might claim to be wanting to stay out of it, but any normal person who wants to stay out of a situation, doesn't accommodate the situation in their home! If the mum wants her there, then she can fund it, she can pay for the food, bills and extra costs associated with having an extra person in her house. Don't make this insanity easy for any of them. If your DD wants her school bag, she can come and get it and she can pay for the transport too. If she wants money for things, she can either ask this other mum or she can get a job. If she wants to play the independent adult, then she can act like an independent adult. Once they no longer have a bad guy (you and your DH) to fight against, I imagine the mum will soon get sick of having to parent and fund someone else's child.
If it's helpful, keep talking to us. We have your back.