A lot of people don't seem to understand that school is a Lord of the flies situation. Teachers can't get involved, and if they do they often just punish both students because they are told the primary concern is to protect the school from liability, not to defend students from bullying. Students need to be able to stand up for themselves, because as it stands now schools are fundamentally incapable of protecting children from bullies.

I don't know what the path is to start mending the relationship with your wife, but I think you did the right thing by your children.

Maybe admit fault, that you shouldn't have gone rogue and endeavor to include her in the future. Suggest enrolling the kids in a martial art so they can defend themselves but learn in a responsible and constructive way how and when to use physical force.