If you are married then the father has equal parental rights, and you need to be aware that in most NGN tribes the children are held to 'belong' to the father. I doubt whether the BDHC would be able to help you under these circumstances, but it is always worth a try, after all there is a child involved.
With you being a NGNwife on an AR CERPAC I would hazard a guess that immigration officials would be more likely to overlook the fact that your son does not have one in his British passport. Of course national pride may lead an officer to raise the question as to why he does not have a NGN passport.
As far as NGN goes, I do not know whether in the case of one parent travelling with a child he or she must produce a consent form authorizing him or her to depart the country. Is there a supportive NGNwife group in your area? Maybe someone in that group you can trust sufficiently to find out about this discreetly?
On the off chance that things are not yet that bad: Could you not just say you were going on a holiday or to see a dying family member that is close to both you and your son?
With regard to your destination SA: Are you aware that since 01.10.2014, in the case of one parent travelling with a child, he or she must produce an unabridged birth certificate and consent in the form of an affidavit from the other parent registered as a parent on the birth certificate of the child authorising him or her to enter into or depart from the Republic with the child he or she is travelling with? In practice this may mean that you may have to produce these documents at check-in. You need to find out about this.
BTW, I have worked in the region and do have more than superficial knowledge and understanding of life outside the expat bubble.
Good luck and let us know how you get on.
C.