I think that if the school isn't a good fit for her then yes move her. However, your subsequent posts seem to suggest that your dd may just be different or less grown up than her peers and that this is causing her difficulties with fitting in to her peer group. If this is the case then she may well have the same issues at another school plus she has the added barrier of making new friends when the friendships are already established.
You talk about the school having a reputation for pastoral and being diverse. That may be true and I think all parents need to be aware that the school cannot solve all their kids problems. Think about what would help your dd in school.
What school could do if it would help
Most schools have pastoral support. However they are over stretched and often deal with kids with self harm, school refusal etc. There won't be an on tap person who can be checking on your dd daily - not based on what you say in your OP anyway. She may get a weekly appointment dependent on the way the school organises this. Some schools have people purely on pastoral; some don't.
They may have some Sn clubs such as lunchtime clubs that she could attend. However, your OP suggests to me that it is not necessarily something she would want.
If she has trouble getting in, they can offer meets and greets and safe spaces such as library passes etc.
Deal with bullying so if people verbally or physically abuse your dd they should act to give consequences and do all they can to keep her safe. The incident at the park is tricky as it sounds a if it happened out of school time and with a pupil from a different school. They could however clamp down on people sharing it. If you know who is sharing it, tell the school and they should contact the parents and speak to the child.
They could change her form group if you think that would help. They could move her out of groups where she has difficulty with certain students depending on her ability and the way the timetable is organised.
What they can't do - they can't make friends for your dd. They can't prevent the fall outs that happen. They can offer clubs and other places where she can meet new people and I would encourage that.
I hope things improve. It may just take longer for her to find her feet. The difference between primary and secondary is vast. Some kids are just not ready in Y7. I used to work in a middle school setting and that really was great for kids like your dd who needed more time before making the transition to secondary.