The fact is sd is happy with all this as it means she gets to spend a full day with dp every week while we are left at home
She isnt going to think about you at home, she is thinking about her Daddy! Your post sounds like you think she is manipulative and delberately trying to freeze you and your DC out of your DHs lives. You have him all week, she gets him one day a week, thats all she sees. She doesnt see that its his one day off, she sees it that you get him most of the time and your dd gets her daddy all the time and she doesnt. Thats probably why she accused your dd, she is jealous that her sister gets daddy when she doesnt. Doesnt make it right, but makes it understandable.
She doesnt know what she did was wrong because no one has told her. Her mother backed her up even after she admitted it was lies, so she has no moral compass to guide her. The problem isnt the child, but the mother and as long as the child is under the age of majority, you will have to deal with the mother. As sickening as it is, you have to suck that up and make the time your SD has with you as stable as possible. This time away from your home has probably hurt her an awful lot, so I think she will be feeling some regret for what she did now, even if only from a selfish point of view.
She is only 11, a child not an adult. She has been brought up by someone who is clearly a bit unbalanced in terms of her ex, and therefore as far as she is concerned, chucking a tantrum and telling lies to get her own way is fine. Its what her mother and the family do, so its normal to her.
She is still a child, and a very confused child at that by the sound of it. Yes it must be hard to know what you and your dd were on the receiving end of something you didnt deserve but chances are she didnt realise the effect it had on you. Expecting an apology for something she doesnt understand is a bit much tbh.
Could you talk to your DP and suggest that she comes over and you and he have a talk with her on her own (after your DC is in bed?) to let her know that what she did upset you very much and that it wasnt acceptable. Say that you feel that it would be good of her to apologise to your dd and see what happens, chances are she will apologise to you too. Saying to your DP that she cant come into his home until she apologises is wrong. It is her home too as he is her father and short of violence or abuse of some sort, nothing should stop her being allowed to be there.
Dont let this become a problem between you and your DH as that sounds like exactly what his ex wants.