She sounds horrible and I can see that you want the best for your husband child. You know that he has the ability to do well in school if he could get away from the bullies and a poor school.
I think that you and your husband need to have a chat with her when her child is not their.
Ask her is it more important that X is bullied, called names and stays in a rough school to suit her?
My feeling is that she worried about what her neighbours and friends think if let's him go to a better school.
I know people like her who may not have been good in school. Had parents with poor education and had no interest in their own kids education. Then she may have grown up in that area and gone to a poor school like her son and does not realise the benefit of a different/better or so called posh school.
Ask her has she any friends whose kids have left that school? Ask her what are they doing now? Are they on the dole, have a few kids with a few different father's or are they in jobs, doing apprenticeship or in university?
Your husband needs to tell her to grow up and stand up to her. He needs to tell her that he is not going to sit by when their son is being bullied in a rough school. That he wants him to change schools to get away from the bullies and to have more subject choices and better teachers.
Tell her he did what she wanted a few years ago but he is no longer putting up with her calling the shoots. Tell her that her son wants to change schools to a better one.
She can let this happen and not be running down the private/grammar school because she does not want him going their.
If she does not do this he has no other choice but to go to court and tell the judge what she is doing and how it's impacting on her son's education.
You could say to her also that you and X ( your husband) have always been fair about child maintenance but you feel that she is getting a bit much so you might look into this also.
My feeling is that when she realises that you going to give her less money and bring her to court she might see things differently.