I can relate to your situation. My daughter is almost 11, and I face a similar situation where I don't agree with her mother on her upbringing, even though not quite as extreme as in your case.
The first thing you must do, is to make an appointment with a child psychologist. The courts makes orders which are in the best interest of the child. The mental abuse is not in the best interest of the child. It is even your partner's DUTY to do something about the mental abuse and try to stop it. There have been court cases where a mother, who did not do anything to stop verbal abuse, was herself convicted of child abuse. So it is very important to get the child to a child psychologist. The child psychologist will give you guidance as to what might be in the best interest of the child and will also help the child to understand why she is getting mixed messages from the adults.
Secondly, the father does not need to sign any divorce papers. The attorney should simply place the matter on the roll and let the law takes it course from there. If he defends the divorce, then it is going to take a long time, but at least he can't stop it.
Thirdly, your attorney should apply for a Rule 43 order for interim maintenance pending the finalisation of the divorce, together with custody of the child, and only supervised visitation by the father until the Family Advocate has finalised its report and until he has made adequate provision for the child to sleep over.
Fourthly, you should also apply for a family violence interdict. Violence is not limited to physical violence, but also emotional and financial bullying. In the family violence interdict you can prevent him from bad mouthing you or your partner, swearing at you, telling lies, and forcing him to pay the school fees (but the latter can also be addressed in the Rule 43 order) and refund the pocket money and costs of the cell phone.