It's not really about a hair cut, it's about control and the fact that when our kids are away from us we can't control their caretakers or protect the kids from their behavior. And we can't.
Co-parenting is a long game. Fight the big fights, save the haircut fights for bull sessions with your friends.
A big lesson for me to learn when beginning to co-parent was that sometimes we were going to disagree and sometimes our styles and timelines would be different -- and that this is neither abuse nor neglect. One thing about co-parenting is that you're going to have to miss or share milestones with the other parent, even if you don't like them. You don't have to like it, it just is what it is.