Thank you everybody for your replies - so helpful in gaining some perspective!
To answer some of your questions:
If she's always been "high maintenance" how did than manifest and how was it dealt with? When she was little she was always very emotional, very upset at nursery and school drop offs - we always did what we could. I tried to encourage her to get involved in physical activities (gymnastics, swimming etc) and also confidence building things, like drama club. She was very creative so my DP would spend a lot of time drawing with her. We also spoke to her many times and tried to help her with her emotions, and her primary school also helped speak with her.
Reason for the move - It was a big move away from London and closer to my DP parents. Sadly both have passed away in the past year, and whilst this could be part of the behavioural problem, my DD appears quite cold and unempathetic to my DP about his huge loss. This is confusing us at the moment.
The bullying and did I get help at the time - yes, I spoke to and visited the school many times, and arranged for my DD to have a course of counselling. She was taught some techniques to help her manage the situation and her emotions.
"I hear this rubbish at work so often from parents who don't want to parent and think schools will do the job for them." - Interesting comment, my mum is a retired secondary school teacher and says the same. She has said this to me for a long time, so I've always tried to instil the right behaviour and not leave it to teachers to parent for me.
Have you asked her how she would feel if someone spoke to her the way she is speaking to you? Yes we have asked this question, and she just shrugs.
Have you asked her if she would be hurt especially if it were someone she loved? As above, we have asked this question, and she just shrugs.
How is she behaviour wise at school and with other adults? Generally fine at school, and rarely gets into trouble. She is doing well academically. With other adults, it is starting to seep through, especially with my wider family.
Can you maybe see if she would engage with counselling or any mental health support, it sounds like she's struggling. Has the bullying stopped? Does she have friends and hobbies? She doesn't want to do any counselling, doesn't see it as helpful. Yes I think the bullying has stopped, although there was an incident with a boy sending inappropriate messages to her on snapchat which I have told the school about and reported it to their safeguarding team. She does have friends now, which I'm really pleased about, and its what she desperately wanted.