She needs to call the cops and file a report IMMEDIATELY at any sign of abuse. The judges don’t give a crap about pictures or stories- they only care about convictions of child abuse. I am sorry for the situation she is in- I am only cynical, because I went through a nasty divorce with courts who didn’t care about an abusive ex with a skank girlfriend (now wife).
I am no expert in such situations, they all seem so sad.
But I do know if the police are called and there is a mark on the kids, there tends to be more active interference. This may unfortunately have the effect of your daughter suffering also, but when cops come to a house it puts different wheels in motion than just the court.
I don’t know if that’s a good or bad idea. Just throwing it out.
Also, if the child is in the father’s care, and she goes to the school nurse telling them Daddy hit her, that’s another layer of legality.