She is 11. When they are 11 attention-good or bad-is the name of the game. 
 

Don't stoop to her level. Ignore it. Keep doing what you were doing. Just add selfies of you doing fun things together. Photos are hard to deny! Play smarter not harder. 

She is 11. Pretty much how kids this age act.

What bothers me here is that he believes his ex over you, and that he "doesn't know what to believe" when you have told him what happened. He should be listening to you, not a child and certainly not his ex! 

Hey, I can absolutely relate to your post. It is a shame that we are made out to be the "bad" guy when they are in denial on what their child is doing and the lies, but their child could never lie, so it just be us lying instead. Maybe this happened to you also, but if you continue to state that you don't lie and it never happened and because of how defensive you're getting over a lie, which hurts the feelings, but they don't understand why it would hurt the feelings, they will tell you that the only reason why you're taking this personal is because you don't even want to be around or even like my kids. I was even asked for a temporary separation from my own house because of the stuff that bio kid came up with to his Mom and she believed it. I am always scared when they are by themselves because I don't know what mood to expect trom Mom after spending time with him because it happens all the time she each time, I've dont something "insane" or my favorite... "how could you". I'm going to tell you that if they tell you they are going to get better, do not listen to them because thwy won't. You'll have a good few days, and then bam, you're left in the dark once again, but you can't say anything because THEY ARENT YOUR KIDS". 

The issue here isn't the lying from SD. Yes, that is A problem, but one that happens with many kids when their parents DON'T reinforce that lying is bad.

SD lying is a problem for BM and DH. The problem for YOU is your DH not believing you but believing his ex, I'm assuming as a proxy for beliving his daughter.

What you have to do now is stick with what you said. Don't want to be alone with SD? Tell your DH you won't be. On days he leaves her with you, either wake up earlier than him and not be there OR call him and tell him he either comes and gets SD or you'll be dropping her back off at BM's or grandma's or wherever.

"DH, I am frustrated that SD lied about me, but she's a kid who needs parented out of that behavior. What I'm FURIOUS by is you thinking I would do any harm to your child AND THEN LEAVING HER IN MY CARE. It's you BELIEVING your XW and daughter ahead of me when you WITNESS how much time I spend with her! And you didn't ever actually believe me - you just accepted that SD was lying after you became her target and it didn't feel good.

DH, this is a failure on YOUR part, one to be a good father who teaches his children that lying has consequences AND who leaves his children with someone who apparently causes harm to them, AND to be a decent husband because instead of talking to me to find out what happened, you automatically assumed that I MUST be lying. Your wife. The woman who helps RAISE YOUR CHILDREN that aren't even her own, and who babysits them for you without so much as a thank you.

You've now lost the privilege of me being available for your parenting needs. You believed that I would do harm since you believed I DID do harm, and I am not putting myself at risk of being lied about again and not believed. SD does not stay with me alone until she learns from her consequences and gains by my trust. If you leave her here with me, I will give you one chance to come home and get her or I will drop her off with BM or grandma. I am also going to talk to SD about lying about me and how that made me feel. SD may have lied, but you're the one who effed up!"

With SD, I'd tell her, in front of DH, how disappointed and hurt you are that she lied about you, and how that has really made you lose trust in her. You can explain the consequences of losing trust in her - that she'll be getting a different babysitter, she won't be able to stay at the house when DH isn't available, etc. Or you can leave it at you're disappointed. It depends on how SD reacts.

Don't let this slide with DH. This is a pretty mundane eff up for an 11 year old. She doesn't need to be met with fire and fury. DH, though? That is a man who needs it laid out in front of him that his showing of lack of trust has massive consequences to his marriage.

It's not normal 11-year-old behavior. It is common behavior for kids whose parents believe whatever they say about the other parent's home. She knows she can get sympathy from BM or DH or grandma for being the poor ignored victim (which exists only in her head). And no one calls her on it. 
 

You can't control any of that. But you can decide not to put yourself in a position of being lied about, especially if she escalates her "poor me" narrative into stories about you and DH abusing her. Let DH know that you will not be alone with her and step back from "doing everything for her". Don't be mean to her - just don't be as involved either. 
 

My SS20 played this game skillfully and is now a very practiced liar and victim.

This is hard, and I feel for you. The biggest problem you have is your DH's attitude and the fact that he believes SD and BM (or at least says that he does) over you. Particularly given that he should definitely know better based on what's happening right in front of him. That's something I would nip in the bud ASAP, though I don't really have advice on how to do that because I've not experienced that.

As for SD lying about you, I have experienced that and I know how painful it is. I've known my SDs for 12 years and the amount of time, love, energy, thoughtfulness, and money that I've poured into them is astronomical. For years we had a very loving relationship, then in the early teen years the lies started. "Felicity isn't nice to us, she doesn't love us, she never does anything for us, she doesn't want us around...." and on and on. It BROKE my heart in the worst possible way. Mainly because I could not understand how they could possibly say things that were so obviously untrue. Everyone in both my and DH's families were shocked. People tried to intervene, including my MIL, FIL, BIL, SIL, and of course DH. They begged them to recognize how much I loved them. And in the moment, my SDs would always acknowledge that they knew how I felt. And they'd write me flowery letters and social media posts, and then within a really short period of time they'd "flip" again and be calling DH ranting about how much I hated them. We went round and round like this for years. I would swear to disengage in order to protect myseld emotionally, but I couldn't stand to see their needs unmet or for them to feel left out from anything good, and so I'd always end up doing things for them again. 

My biggest regret in life is that I didn't disengage from them sooner. Really and truly disengage - like I wish I'd just treated them like a friends' kids. Politely, maybe even affectionately, but not doing anything for them that actually cost me time, money, or emotional energy. It would have saved me so much heartbreak in the long term. Right now we're in an awful period where they not only hate me, but they are deeply jealous of my DD and their disdain still hurts me so much. I'm working on it, and I wish that it didn't bother me. But I don't know anyone who can love and care for children for years and not end up feeling hurt if those children eventually reject them without provocation. 

So my long-winded advice is to protect your heart. An 11 year old child telling blatant lies about you is not likely to stop so long as she's being rewarded for this behavior by one or both parents. In my experience, there is no amount of love or emotional labor you can do to prove to her that you care for her when she's getting another emssage from her BM. So why bother?