why invite this child to such an important event knowing how he will behave?

Since the mother refuses to acknowledge what happened, the mother of the birthday boy should stop inviting her child to any more children's events. If the other mother confronts her, just say that her first priority has to be her own child, and she is not going to risk his special events being ruined again.

If that ruins the relationship, so be it. The first responsibility of a parent is to protect their child. The second is to teach manners and morals so that their child doesn't become the type of brat that needs to be banned from events.

BTW, there IS a chance that the badly behaved child has a real medical condition that needs treatment. So if I were the mother of the birthday boy, I'd mention to her that if there is an issue, diagnosis and treatment would help the child. Even if this meets with anger, denial, etc., it's worth doing if it might help the child...