Look, EK's rules are pretty clear and they don't have reason to make an exception for you. If you ask nicely, maybe. But the rules are simple. F pax only. No guests, even if you're Gold, unless perhaps your uncle is the sheikh.

You could have easily gone to the J lounge with your son. For the purposes of your ticket, he is a guest. And don't compare having no status on EK to being a Gold or above on BA. You're going to get little sympathy on here, I'm afraid.