I feel now that I can comment without going over the line as you have opened up very well. Please remember though that I am taking what you have told me and going from that.
Don't ever give up on a child....!!!! Period. That is what the problem might be with the mother. It sounds like she is too angry at your husband for some reason to be fair with her own daughter and give her the chance to grow up knowing her father in a positive way. That is the most selfish thing a parent can do. I have had all sorts of experiences with previous girlfriends and friends who use their kids as pawns. They are more often than not very selfish people that never consider the problems they cause for others. Your stepchild will not benefit from you or your husband giving up on her at anytime. Especially now if the mother has an agenda to discredit the father in the daughters eyes.
Your stepchild is acting out because she probably thinks her father left her for you (her mothers doing probably) or doesn't love her anymore or did something bad to her mother.
There are a few things your family will have to do.
1. The father is going to have to spend time alone with his daughter when his visiting times are on for a few months. That means you and your newborn will not be invited under any circumstances. This will rekindle the lost relationship they once had. You and your babies existance should never be brought up until the daughter asks. This is harsh but it will work. The father should not badmouth her mom under any circumstance, even to defend himself. This is the biggest no no a parent can do. It will have the opposite affect and will only confuse the daughter into believing her mother more. He should not say anything but everytime he drops his daughter off, before her mother is visible, he should tell her she will always be his little girl and that he loves her so much and she should always come to him for any advice she needs, no matter what it is. This will get her thinking good things again and she will slowly start to see the real picture again. She will also realise that her dad is not the bad man. I would also suggest that the father invites the mother after a few times knowing she will decline. This will start to show the daughter that her father is not the bad guy her mother is portraying him out to be.
2. After a few months, and only if the daughter is interested can you be introduced again into the relationship. By then, your husband will have gained her confidence and trust again to be able to make her understand that he hasn't left her for anyone. You will need to be patient and trusting that she will come around eventualy. There should never be a bad word said about her mother by anyone as this will only make her want to defend her mom. Let nature take its course and her mom will eventually show her true colours one too many times.
3. When the daughter starts being interested in your lives again, always invite her to join you whenever wherever you go to show her she is as much a part of the family as your newborn. She is old enough to be honest with and communication all around will only benefit her. I have been there as a child in the same position and then as the other guy in a few relationships with mothers and children. I was always honest with the children without telling them the truth about their parents and found they were never threatened when I was with their mom. I also took it on the chin when a few would tell me they wanted their parents back together. This is part of being a kid. I actually admired one of the kids that said that to me because she felt close enough to me to let me know her true feelings. We were very close for a long time but when her mother and I were on our way out, her true loyalty was with her mom. That is how it should be. Whether she might have been spiteful at the time is not important. I always allowed her to speak her mind as long as she was polite and thought about what she was saying before she said it. She was 8 years old then.
4. You should rent some movies with the same problems she is facing and let her watch them because if she sees it happening to others, she might realise what it looks like on the other side.
5. Protect yourselves legally so the mother cannot take her away for good. I mean setup a hidden camera on those face to face occasions with the mother and record the telephone conversations. Who cares if it doesn't stand up in court. The proof will be there for all involved.
I hope this sheds some light for your family. It is really hard for kids to have to adjust and sometimes without wanting to, take sides. Don't ever ask her to choose between her parents. It's not fair.
Best of Luck
David, Sydney, 8 mth baby