I have to be extremely overheated before I sweat at all, and even then I don't break a heavy enough sweat to help me cool off. All it does is make the surface of my skin a little bit damp and salty which then burns a lot. I usually only sweat at my thighs and my upper back.
I didn't start to sweat at all until my very late 20's and I have lamellar ichthyosis. It is very normal for folks with LI to not sweat at all.
You might wish to look into cooling vests so that your child can play outdoors even when it's hot and not overheat so easily.
I am female, and was born in 1972 with Lamellar Ichthyosis.