I think it’s great that the two of them are with your DB - I hope they stay there for ages so you can get more observation reports!
Your DH needs to understand that the only person who can begin to fix the breakdown in the relationship between him and his daughter is him.
He can’t outsource it to you under the catch-all ‘wifework’ banner, it’s like his personal hygiene - only he can do it (all you can do is ensure there is toothpaste and shower gel in the bathroom ready for him to use).
As for the mother, I wouldn’t do anymore than a casual check in as to whether anything has changed since your previous contact (eg ‘Just checking in with you as I realise a whole extra person in the family home is quite a big change and I wouldn’t want you to think we’re taking you for granted nor that DD isn’t welcome to move home whenever she’s ready’) but no offers or enquiries about financials or future plans. If she says everything is fine say you will check in with her again in a month/2 months/before exams start whatever feels right to you and then stick to it.
Your aim should be no more than creating a communication channel, so that DD & X can’t exaggerate or invent problems without you and the Other Mother having an avenue to double check if anything escalates.
Also, OM needs to feel like she can simply send DD home if she’s had enough of an extra mouth to feed - none of us would feel able to do that if we truly thought the teen’s home environment was abusive or damaging. Phoning up and getting shirty will reinforce the false notion that DD is unsafe in the family home, whereas a calm and undemanding check in will demonstrate otherwise.
I would write a simple breezy text and I would immediately screenshot the text and send it to DD with a ‘Hey [pet name] I sent this check-in text to X’s mum, just letting you know. I love you, mum xx’
If DD protests you can say you value honesty and transparency between you and DD and thus wouldn’t communicate with X’s mum behind DD’s back but as a responsible parent you had to check that she still had a stable roof over her head.
Pretty much all of us have a natural inclination towards wanting to be seen as the good guys, who wouldn’t rather be The Fairy Godmother instead of The Wicked Stepmother?
If you give The Other Mother the benefit of the doubt and imagine that she is just trying to do the right thing for a kid who has been framed as vulnerable/misunderstood at home, then the way to approach her is by gentle, indirect reassurance that you are not The Wicked Stepmother and DD is more Dick Whittington (setting out on an adult adventure) than Cinderella (neglected and forced into slavery within the home)!
Teenagers paint everyone and everything with a massively unsubtle big brush with no nuance. Let some nuance back in (and tell DH he’s behaving in the same way as the teens by making the other family out to be pure panto villains when the truth is likely more complicated).
Is your family therapy coming up anytime soon? Your DH really needs someone outside of the family system to tell him to stop outsourcing to you and start communicating directly with his daughter. When did he last text her? Did he ever apologise to her for cancelling on her?
(I’m still pretty convinced her subsequent cancellation on him was a revenge response, although perhaps subconscious on her part).