God, this is sad. Everything you're saying about your stepdaughter indicates that she is anxious around bedtime- and seeing as she stays with you only one night out of seven, it is pretty understandable. Not that this is anyone's fault, but she's coming into a home with 2 other children, both of which count that house as their primary home- That's probably quite difficult for her. She needs a hug, smiles, reassurance. She needs to know that you all still love, value and notice her even though there is a new baby.
I realise you're tired with such a young child, but you write about her as if she's an annoyance. I'm sure you do your best to disguise that, but it's very possible that she's picked up on it. She's just a scared and vulnerable little girl.
What if your DH takes her to bed and takes his time with her? Say he'll read her a story and stay with her until she sleeps- he can read a book or something while she dozes off? Be honest- tell her he's going to go once she's properly asleep, but if she wakes up again, he will not be far away and can always come for a hug. Maybe say that if she wakes up at night, she can put the lamp on and read a book until she feels sleepy again.
This situation needs kindness. She deserves kindness.
BTW- In the same way as you think attachment parenting is wrong, I think it's very wrong to deny a child comfort when he/she is afraid and alone. We have different styles of parenting, and that's fine- but you have no right to expect this child's mother to change her parenting style because you feel differently to her.