DD2 had a so-called-friend like this.
She would stop talking to her for no reason at all, then after a while, beg to make up again. After a while, DD got tired of the routine, so said she wasn't interested in being friends anymore. So the girl started to get other girls, and then teachers involved, saying how unfair DD was, and how cruel not to make up after an argument (which, of course, was not an argument, at all, but a whim of this girl). She tried to isolate DD from other friends, by standing in front of DD when she was talking to someone, or calling them over about something apparently important. Etc, etc, etc.
In the end, it took the girl blaming DD for spreading a rumour about her, which she hadn't done, for the girl's mother to get involved. She telephoned me about my DD's behaviour, and I just let rip about her DD in the end. The mother could not imagine that her little darling could ever have been so nasty, but at least I managed to convince her that whoever was to blame, it was not a good friendship, and perhaps it shouldn't be encouraged.
I can't say it's been completely plain sailing since then, but she has backed off a lot. She doesn't play these games with DD anymore, although they are in the same friendship group at school.
I agree it has to come from your DD, but do encourage her to see the difference between forgiveness and out and out doormattery.