I applied for my open work visa, and submitted my wife dependent work visa and my child student visa applications at the same time. However, I included cover letters to make things clear why we only submit this and that papers but not some other stuffs per requirements in INZ form. INZ called me, said that it would be better that I applied my work visa first, then apply visas for my wife and child later when my visa has been approved. A moment later they called again and said it was OK to apply all three, but we should be fully aware that the success of my wife and child visa would depend on the success of my own visa, I said OK then they proceeded visas for our whole family within 04 working days.
So if you still worry, maybe you should apply student visa for yourself first, then apply for your son. Perhaps in that way INZ would give out the student visa with domestic status (but not visitor visa) for your son more easily.