Not quite the same situation but my DD was targeted by someone in her class in Y7. It was definitely bordering on bullying so when I found out about it, I emailed the form tutor and head of Y7 with a very factual account of the incidents I was aware of. I requested that DD be immediately moved away from the culprit in every subject and ideally, for DD to be at the very back of the classroom to minimise opportunities for the culprit to do things behind DD's back. I also asked for the whole class to receive a reminder of the school's behaviour policy and specifically, what it said about bystander bullying.
Within 24 hours of sending that email, all the students involved (witnesses, victims and the culprit plus her sidekicks) had been spoken to. Seating arrangements were changed as requested and the school handled it all in such a way that none of the students (DD included) were ever aware it was a parent that had alerted them to it.
For about a year, things were quiet before escalating. The culprit threatened to throw DD under a bus and later dragged her by her hair into a stairwell and beat her up. She was watched, encouraged and filmed by peers. DD's glasses were removed and used to beat her around the face before being dashed to the floor. DD was a lot smaller than the culprit and made no effort to fight back.
The teachers who intervened did so because they heard a disturbance. Everyone who dealt with DD in the immediate aftermath was kind and looked after her but mistakes were made when it came to dealing with the culprit. Whilst she was initially suspended, the school did not stick to its own behaviour policy which led to the opportunity for further bullying behaviour by the culprit and her allies. Throughout this, particular teachers were genuinely amazing in their support of DD but the reluctance of the headteacher to follow the behaviour policy undermined this. When the culprit returned to school, she was moved into a different form and out of any classes where she had previously been with DD. She was also supervised at break times for at least a couple of weeks.
Because of the mistakes made, and because I knew from another source that the culprit was still boasting about it online, I felt she hadn't understood that if the behaviour policy had been followed, she would have been permanently excluded and so still had the upper hand over DD. The school was not willing to revisit its sanction and I was advised to report the assault to the police. I had hoped to avoid this but ultimately it felt like the only way DD could demonstrate that messing with her has consequences. DD had to give a formal statement and the culprit was interviewed at the police station (caution plus 3 interview). She ended up with a community resolution order and since the interview, has steered clear of DD.
The process took a long time for a number of reasons and alongside it, I submitted a formal complaint to the school about aspects of its handling of the initial incident but also about the ways in which it had failed to safeguard DD. I made sure to mention by name the teachers who had supported DD throughout and explained I felt I had no choice but to complain at the highest level to ensure the safeguarding failures were addressed. I received a very prompt response to my complaint (sent to the executive headteacher and the chair of governors), met with the EH at 7am a day or two later and was finally reassured my concerns would be looked into. Again, this process took some time but it was thorough and resulted in the film footage of the assault being passed to the police. The EH said watching the footage was horrific and advised me never to do so if I were to come across it. Eleven of my twelve complaints were officially upheld with the remaining one being verbally supported. The school has made changes to a number of practices as a direct result and DD feels safe at school.
I've explained all that because I hope it will help you to be more assertive and demanding on your DD's behalf @KarenNotAKaren . Look at the behaviour policy and use that to write a letter of formal complaint to the headteacher and chair of governors, stating that your DD is being targeted at school by X including deliberately pushing her over on the ice, pinching and barging. Explain that you were one of the signatories of the group letter but that you understand that may not have given the school a clear picture of what your DD is being subjected to. Mention the steps you have taken already e.g. arranging a meeting with the teacher etc. but that you feel the school is not keeping your DD safe. Having read the behaviour policy, raise anything from that which should have been done but hasn't e.g. a move to a parallel class, break times with a different key stage etc. Finish by saying you want to be able to trust the school to deal with the situation so your DD cannot be hurt again at school but that you have been advised to report the ongoing physical assaults to the police.