Been there. Well, can't fully blame him for moving out if you told him it's the end of the road- I had to swallow my threat once too. Time to have a quiet talk with him and see if you two can reconnect and maybe SD can stay at BM for a week to allow you to reconnect. I have a SD11 just like that. We strengthened our relationship and made it clear to her that she is not breaking us up-ever. I since backed off, I am just the nanny, I keep her fed and alive, minimal care, and see how it goes. Basically, re-establishing boundaries for my sanity. I feel for you.

I got SD11 a phone- she never calls or answers when her father is looking for her, uses it for cyber bullying and sexting. I took her phone away. I bought things for her-never a thank you and always an ungrateful demeanor like I owe her. I take her places-same. I try to talk to her- eye rolling and negativity. She is unapproachable. I want to set up hair appointments or lessons off her, it's gonna be a no show, so why waste the money or time? My GF calls me the Bad Nanny because I have less rights or authority than a good nanny would have. So, cheers to being the nanny!

I have a lot of experience with this, my SD is now going on 16 and she lives with us full time. No BM in picture and she has always behaved the way that you describe and her father does nothing or makes excuses for her out of guilt every time.
So I have learned to remove myself from doing anything that requires me to wait for her or to schedule anything with her. I had the same thing happen with a hair appointment years ago and never made another one. I take my own car when we go on family outings or if we have to be somewhere on time. That way I know I will be on time and I can leave if stuff gets weird. I basically was disengaging before I even knew what that meant but with the help of people on this site I have learned what it means to disengage and I have gotten some support over it.
I suggest that you slowly start to disengage. This does not mean that you do not have to be involved in this child's life, you just always assume what the worst possible outcome could be when making plans with her and you plan around it. Her dad can take her to get her hair cut. You can conveniently be busy or have some errands to do if you have anything planned that has to do with going anywhere with them and always just say to them that you do not want to hold them up with your errands so you will take your own car and meet them at the designated spot, or you can feign having some errand after the fact. This way he can wait for her to show or not show and it is not your problem. I know that this is not exactly what you posted about but I have found that not making plans to do something punctual with a teenage girl that behaves like this has saved me a lot of headaches. It has also forced her father to deal with it.
As far as the excuses go, ignore them. Do not comment on your DHs precious one's behavior as I have found that he will tire of hearing about it because if he is that full of excuses for her this will be a hard habit for him to break.
When he starts getting inconvenienced by her tardiness and inconsideration only then will he address it. At least in my situation this was the case.
Good luck, I am glad mine is not 13 anymore and now is 16 but we are now faced with new challenges even though she has matured slightly.
I have found a lot of support here on this site and I hope that you do too. You are not alone.