She hates her body - Pretty normal.
She hates having periods - Don't we all?
She feels constantly uncomfortable in herself - Still exploring the world, pretty normal.
She doesn’t want to grow up - Did any of us want to grow up?
She doesn’t want her breasts to get bigger - Makes sense if she is still exploring who she is, also, make sure she's not in pain, that can be a reason for some. I had friends at school who got really bad chest pain during puberty.
She doesn’t want men or boys to look at her sexually. She doesn’t know if she’s attracted to boys or girls - Sexuality can take time to develop, give her space, and make sure she feels safe at home to explore.
Being a boy feels ‘safer’. - Goes without saying sadly.
She wants to cut her hair and wear trousers for school. - Fits in with the above, but also, trousers for school is possibly something to make her feel safe at school.
She wants to use a boys name. - Okay, try it, no harm in trying a name around the house, if it doesn't feel right for her, then she'll learn more about herself!
She doesn’t want to see a doctor or a therapist about any of this. Nor does she want the rest of the family or teachers at school to know. - Well there's no need for anyone else to know she has therapy, but I feel it's very important that you get her to see a therapist, there's no shame in it, I have seen many over the years. If it bothers her when talking to friends, she could tell them it's a book club she goes to.
She said she felt sure she was trans until I spoke to her and now she’s not so sure. - Needs room to explore.
There are some things here that I would say need investigating by a trained professional. You're really going to have to press for the therapy.