There is no way in the world that I would go on a trip of that duration. When a young girl is exhibiting self-destructive behavior, the #1 priority becomes stability. I am sure her therapist is telling you this, but I'm not sure if you're hearing it. There has been too much upheaval in her life already, with her dad leaving, constantly changing schools, and some weird trip of indiscriminate purpose halfway across the world. She is telling you through her behavior that something is wrong, and cutting isn't a problem that's fixed with a month or two of therapy. Concentrate on fixing that before you move yourself one foot.
I might see an argument for taking her on a two-week vacation - that would be absolutely it. A 2-3 month trip would be more traumatizing than a 6-month; at least with six, she has time to make friends and frantically try to settle in. This is not to mention what kind of an impact either curriculum break would have on her education. She wouldn't leave and come back at the same place in terms of course of study.
I know you married young and had babies young. I know that you never got a chance to see the museums of Italy, and the world, really, on your own terms. Given that you sound like a terrific mom, and however painful it may be, you know that her needs have to trump yours at this point.
Sorry, brutal, I know, but that's my $0.02.