I haven't suggested that you are not providing stability.
I'm actually suggesting that the mum is not providing stability, and that your son needs stability above anything else.
You have a golden opportunity here to be the rock for your child while she has boyfriend after boyfriend. That must be very unsettling for your son.
I think making an issue of the hair with the child himself by taking him to have it cut would have a harmful impact on your relationship with him. Let it go for now.
It's very possible the school will have something to say about the hairstyle anyway.
Yes, for the future, if you have joint custody, this sort of dramatic change to the child's appearance should be discussed and agreed upon.
Is your custody/ visitation arrangement court ordered? Or informal?
If it's informal, I'd advise you to go through mediation and get an agreement that the family court will sign off on, with both parents bound by its terms, with the best interests of the child endeavor.
Mediation can also help separated parents come up with an agreement to discuss issues of all kinds or even provide a safe place tondiscuss the issues. As long as both parties are reasonable people prepared to put the best interests of the child first, it can work well. If one party is playing games, or if there is a lot of friction, then coparenting will be a very hard endeavour.