It is no business of my employers where my son goes to school.
Depends on what your employers social media policy is. Many now have codes of conduct where you can't say certain things even if private because they may be deemed as bringing the employer into disrepute. Certainly NHS and other public sector staff are particularly strict about this.
I have to say, how did a private post come to the attention of a head teacher. Tbh I'd be more concerned about that, rather than seeking to make a complaint about a GDPR breech which now looks exceptionally vindictive. If you do pursue that course of action why do you think that's in your child's best interests? You force a situation where you are either trying to actively force the head out and/or put your own child in a middle of a huge row between you and staff. The breakdown in communication really makes it almost impossible for your child to continue in that school without it being a problem. This is a scenario that could still be avoided.
As I say, I'd be examining whether my post really was set to private or not (it suggests your level of privacy isn't right) or looking at my friends list. Is it really private if you have hundreds of acquaintances familiar with the school (because they are other parents or local and know where your child goes to school even if you don't name the school).
The fact is that the head will be easily identifiable in your online community to anyone who knows you. You don't have to name the school for it to be obvious who you mean.
If you had a problem with the head, it would be wise to actually deal with the issue rather than ranting about it on Facebook.
You've been caught out here, because you think you have the right to potentially liable someone - which would definitely fall under disrepute policies if your employer has one.
I can't say I think the head's approach is necessarily right or fair here.
I do think you are totally going about resolving this in a poor manner that will ultimately backfire on your child though and you don't seem to wish to be responsible for what you post on social media.
Realistically nothing on social media is private, despite privacy settings especially if you have a large number of followers. Youve potentially publicly said this to a couple of hundred people familiar with the head and well aware of which school it is. You may wish to reflect on this and your own behaviour.
FFS GDPR breech? That should be the least of your concerns at this point.