From mom's perspective, the use of the terms "bullying" and "manipulative" numerous times, may put her defenses up. NOT THAT SHE ISN'T.... Because obviously there are some serious things going on. But I don't know if there are other words to use? Unless of course, mom has been using the terms herself. I just would hate for your message (which is very well worded and delivered otherwise) to get lost because mom gets a little pissy over it.
Is this girl the oldest in care? She may need to be in a program that has older children around. I had a child like this last year, who always had to be "the best" and "the winner" and "the first". I actually think it was because mom and dad praised him TOO much. They never taught him to be a good sport, or how to let others be good at things too. Funny how you can go wrong with being too kind to a child! He is now in a school age care program, and as far as I know, does much better there.
In general, I think that your letter, finished off with what Jazmic suggested, is well done. Fill us in on how it goes!