What would happen, OP, if you did the following:
Had a calm measured discussion with school about seeking extra support, help, counselling, referral etc for your DD.
Called SS and requested help in managing behaviour that is dangerous to your family.
Made an appointment with your GP, urgently.
When your DD returns home tonight, after having all day to run over the events of the morning in her mind, you tell her you are going to have a talk later. Get the other children to bed, business as usual.
You sit down together. You explain to your DD that she is your daughter, and you want what is best for her. You wouldn't allow a bigger, stronger person to assault her in her own home.
Her behaviour today has put you in a position where you have been forced to act. You are willing to support her through this, but she will need to do her part. She will cooperate with the school. She will attend a GP appt with you, and meet with SS. She will agree that in the event of a recurrence, the police will be called, despite your reluctance to involve her with them.
What will happen? Will she kick off in front of her head of year?
Please, just see if you can make a plan, and get on it. Maybe this morning is the wake-up call you both need, the beginning of a huge change in things.