You said it exactly: "she is doing this to get her Dad's love and acceptance." I don't believe it is any more than that.
Your sister's ex is behaving terribly, and hurting his daughter by doing so. Unfortunately, your sister can't do anything about his behavior.
Your sister needs to know that she needs to keep being the same loving, nurturing mother she always was, (while not letting her daughter be a complete brat to her), and her daughter will eventually know who is the better parent. She has to be patient. However, she shouldn't try to "buy" her daughter by giving her necklaces to make her daughter like her. That's phony and will backfire.
Also, her daughter is on the verge of teenhood, and ALL teens, male and female, particularly start to separate from their moms in their journey to independence. That is normal and healthy, and your sister can't take that personally.
Tell your sister this story: I have always been the main parent, my husband mostly lives in his own world and occasionally jokes at the kids or gives them a little lecture or whatever, but I have been the main orchestrating force in their lives in every way, good and bad, and a lot of times I don't seem like such a fun mom.
A few days ago I was watching a TV show with my 16 year old daughter (who has definitely been doing her own separating from mom in the last 3 years) and the topic of the TV show was about mothers leaving the kids to be raised by dad. My daughter and I were discussing the ethics of the subject, and my daughter looked at me and grinned and said, "Can you imagine if we had been raised by Dad?"
That says it all.