Hi, we moved DD at the end of Yr4, not because of bullying, but because we moved house.
DD struggled a bit at first, the worst part was that she was trying to find a new 'BFF' in established friendship groups. She also fell foul of the "You're my new best friend" "I don't like you any more" "Sophie is my new best friend" stuff that I am told is quite normal for ages 8-9-10.
One thing that did help DD was 'fitting in' with the new girls. For eg, at her old school Monster High was 'babyish', as were playing with Bears and trampolines - they were all things she 'gave up' when she was 8/9 (it became One Direction, Pop music, Hey Jessie and iPhone Apps).
Here the girls play with Monster High dolls at break times, DDs birthday 'party' was taking some friends to Build A Bear, she asked for a trampoline for her birthday, despite us giving one away (with her blessing) when we moved and none of the girls are into 1D or trying to turn into teens. They play very simple old fashioned games out of school (although Minecraft is still big) and are much more easy going. She now has a group of friends (4) rather than a BFF and is enjoying being a kid again.
OP You say she finds the other girls babyish and she doesn't like them, well maybe she should try a little harder to fit in - why are they 'babyish', what is it that they do wrong? Is there really no common ground? Can't you push her to make friends rather than a 'best friend'? Does she call the other girls babyish? Is she trying to join established friendship groups or take someone away from one because she wants a BFF? Because none of those things will change in another new school, unless she does. Can you push the notion that a group or gang of friends is better than one BFF, I did that with DD and it improved things no end.