A.V.
Ah, kids. When I was a kid, the farthest I could run was Grandma's house - and she told me to shape up and go home and talk to my mother. I think that your SD is probably a little jealous and is using the tools she can to react and show how angry she is - teens are horrible sometimes when they think things are "unfair". I joked that my SD and DD were in the same phase when one was a teen and the other a toddler. I was actually serious.
Have you considered her father going to the town she lives in with her mom and taking her out to dinner? Spending some one on one time with her and asking, "So, what's the deal here? I hear that you are angry, but I can't do anything if you don't speak to me." I would also encourage him to talk to his ex and ask that she encourage SD to come over as scheduled vs avoiding him when she's angry - which in the teen years can be every single day. Hopefully she is not also feeding the drama for her own benefit.
When my sks were her age, not going was not an option and schedule changes had to be approved on both sides. When they got to be 16 or so, it was more of a loose schedule. (In our case the problem was more that their mom skipped out on visits vs they didn't want to see her.)
He can say, "I understand that you were angry that we went somewhere without you. However, if you do not speak to me, I cannot discuss it with you. Would you like to be included in the future? It was not our time with you, but I can discuss the possibility of a weekend switch with your mom so you can come. I would also like you to show me that you want to be included, and that means coming over on weekends and not just spending time in your room. We can work on this, but you need to meet me in the middle."
I will warn you that around 16 kids get very black and white and will argue the sky is green and the grass is blue. Sometimes it is not worth the fight. But we did try to emphasize with them that they can hold a belief, but they should get all the facts (Dad hates me - why do you think that?)
Good luck. It's tricky when kids are just being "drama teen" (as a fellow SM puts it) but they can go hide somewhere else and "punish" you. It is not always the popular answer but "I will pick you up Friday at 6PM" is different than "do you want to come?" and sometimes they're just fine once it's laid out that they are coming for the weekend vs putting it on them. "Do you want to come?" is sometimes not a question they are ready to handle.
ETA: If he supports her being on FB (isn't 13 too young?) then he should make it a point to go on and look at just her page a few times a week. Because it's important to her. If she feels ignored when she shares something, that is a simple way to say, "Hey, I saw this. Good job." Social media is SO IMPORTANT to some kids. He should consider it an olive branch or meeting her where she wants to communicate.
Also, regarding the younger kids - My SD wanted us to hold of on DD's first Christmas til they got here. I told her and DH no because I didn't want to set the precedent of DD waiting for them and being hostage to their schedules. She's only got one household and should not perpetually wait for the stepkids to arrive. Your DH might want to also point out things SD does with her mom that the other kids do not do, so she doesn't just see it as unfair in her direction. I bet she has done a ton of things with her mom that the other kids did not get to do. The biggest thing here I think is perception and communication and if she won't communicate, he cannot change her perception of fairness.
The problem with the excuse of "needing more time with mom" is that she is with mom more already (why did he give up most of his time?) and it doesn't seem that there is a trade to make up time with Dad - and so it becomes a perpetual circle. She doesn't spend time with Dad, Dad does something without her, she gets angry, and doesn't spend time with him even more. Which is why I suggested making the schedule The Schedule and finding some time to do something one on one and try to break the freeze. It is short sighted to think that a daughter doesn't need her dad or a son his mom. It hurt SS very much that his mom didn't want to spend more time with him. Dads are important and he should make it a point to be involved in her life and not be phased out.