OP, hope you have received some good advice from the posters who seem to know the local schools. Moving, renting, timing - all big considerations.
Please ignore the gloating, sarcastic posters - I am always amazed over how mean some people are with regards to the public /private debate.
I would say that I do believe that your DD will be fine once you are moving forwards with whichever plan you settle on. My DC’s friends / children they come in contact with have a huge variety in ethnic backgrounds /socioeconomic backgrounds (long story but mixture of sports, schools, and hobbies). Some of the friends’ families are incredibly wealthy, two families are making their sons do a huge amount of sports in order to ensure that they are kept away from gang violence (very bad neighbourhood).
The one common factor - and that is the point of this post- between the lovely, grounded children (luckily the ones our DCs are friends with) is parental involvement. Most of the less pleasant children, such as the bullies, the children with anger management issues, etc have either troubled parents or less involved parents.
Your concern shows (I believe) that you are a lovely mum and with your encouragement and support your DD will thrive, both academically and socially.