What is he backing down from? Is it discipline and consequences or not taking her for his time? I think the only way to combat this is consequences for her awful behavior. 

BM threatens your husband with court for what?

Best thing to do is for him to stand up to her, once and for all. BMs love to threaten court but unless they have a valid court case, it's just a threat. Is he going to let her run his life with this threat until his daughter ages out?

Why does your husband allow his daughter to be mean to you and your daughter? Does he not give her consequences? Does he not parent her?

Why doesn't your husband just parent his daughter? She should not be allowed to behave so badly. I bet she doesn't behave like that at school... He's the parent, it's time he got a grip on things.

Please protect your daughter from SD. Your daughter should not have to live with someone who is mean to her. If DH won't start parenting his daughter so her behavior changes, then you need to move out for the sake of your daughter.

dang this kid is 10 years old and rules the house, no way that will happen in my house, any one under my roof will obey my rules, if they do not like it they do not need to visit.

Next visit when SD comes over, Put your foot down, if she is rude, tell her, now that is nasty/rude/unacceptable go to your room. ( Never say please go to your room, immediately you are giving her a chioce to go or not to go, you simply tell, you do not ask)

If SD is at the house and DH is not there, maybe at work, you have full right to refuse looking after her, it is simply, she's not your kid and not your problem, she is not visiting you and your daughter she is visiting her father, he needs to be there or she is not.

Do not leave discipline up Dad, he will never do it, you simply put your foot down, she gets punished for being evil towards you and your daughter, if your DH does not like it, then maybe take a long hard look at your marriage and get out. Sounds harsh but do not waste your time on a man that is unable to teach his 10 year old daughter basic live skills,

She is your SO's child.. and while I can understand her presence impacts you.. your husband needs to be parenting his child.  If his kid is mean.. she gets counseling and consequences for that behavior.  You don't need to parent his child.. and if he won't.. you need to make yourself scarce with your child while she is there.  Force him to have to deal with her.

Your husband should absolutely not give up his visitation. She is 10, you are the adult. HE is her father. Between the 2 of you I am sure you can come up with a parenting plan that holds her accountable for her behavior. 

Have her write apology letters

Have her do yard work when she gets nasty

Have your husband take her out of the house. They can go on a hike or something to get some of that mean energy out.

She doesn't have a choice who you choose as a partner. I'd really consider the effects of this situation on your child and the long term effects on your relationship with your child.  Your kid should always come before someone else's.