There’s a difference between enraged shouting that her behaviour was appalling and name calling. Name calling switches it into abuse territory. Add in the sulking and it happening on multiple occasions and it’s classic abusive behaviour.
I don’t even know if you’re still reading OP, but my eldest teen began to kick off seriously aged 14. Looking back, it was a cry for help because of a bullying father.
At 17/18 I had to watch as he became embroiled in a relationship with a young woman that screwed with his mental health to the point where he was depressed and self-harming.
I could only watch from a distance though, as he’d moved out. But as I watched him struggle in an agonisingly painful relationship, which he seemed unable to leave, I realised that I’d taught him that staying in a relationship, however dysfunctional, was the right thing to do.
I’ve also seen the thread linked earlier about the daughter who is having to withhold her grandchild from her mother because of her father’s bullying behaviour.
These are possible futures others of us have faced as a result of choosing to stay with an abusive man over protecting our children from him.
Sometimes leaving isn’t possible, but if you stay, then you need to protect your daughter. Explain to her that his behaviour towards her is wholly unacceptable, but that for whatever reason (explain as appropriate) you are unable to leave right now. It has to be a genuine reason why you can’t leave and there are some.
But she must be your priority. I made the mistake of thinking that I shouldn’t criticise my husband as it would undermine his parenting. But sometimes if you recognise fully that something is out-and-out wrong, then you shouldn’t stay silent as silence indicates implicit approval.