Sorry i cross posted with you OP.
You are trying to do the right thing,your trying to protect your DD and yourself SS will not get involved and try and remove your DD because of that!
Your husband has been Gaslighting you by threatening you with that.
I've helped 2 women escape abusive marriages with the help of womens aid,one of those husbands had threatened his wife with the same line and she stayed for years because of it.
After she got away the courts stopped him from having any contact with his DC because of his behaviour!
With your DD behaviour alot if not all of her playing upto her Father will be because she can sense the tension bless her,it's sadly quite normal for very young children to do that,it's that way of trying to break the tension that's going on around them.
For your DD if your the one that gets left to cope with her not sleeping or her not doing as she's asked and she messes around more for you than your DH that could be because your the safe parent.It's actually a compliment to how good a Mum you must be and how patient you are with her,it shows that she feels safe with you.
That could be another reason why she's now struggling to sleep at night as well.