I know this is an older post, but perhaps I can offer another recommendation.
Before I go much farther - If you're anticipating the possibility of a fall, most certainly make sure that you'll be using a dynamic rope, as opposed to the more common static (climbing) ropes. If you're unsure, just ask and we can discuss later...
Regarding harnesses - Rock-climbing harnesses may work, but I would certainly fit-test them first...
The Petzl "Ousti"
(pronounced Wee-stee?) is a harness purpose-made for kids. It's probably the best "safest" option, though I can't confirm it's fit on your son...
Petzl OUISTITI Kids Full Body Harness Full body climbing harness for children less than 30 kg Designed for children weighing less than 30 kg, the OUISTITI harness is especially easy to put on and adjust. It does not require a connector to be fastened, and the adjustment system is difficult for a...
www.innermountainoutfitters.com
There seems to be a comparable option from Edelrid as well.
Edelrid Fraggle III Full Body Harness With this padded full body harness even young climbers and little mountaineers weighing up to 40 kg can enjoy a scramble. The stiff construction with the fixed leg loops makes putting the harness on much easier, and thanks to the two Slide Block buckles, the...
www.innermountainoutfitters.com
I CAN confirm that the MTDE Club-Caving harness is nearly infinitely-adjustable. I had my son in one of these at 6, and it would still adjust down smaller. It does require a semi-circular "D-ring" for closure. If you went this route, I would also pair with a chest-harness too, though it doesn't need to be substantial.
Club Caving Harness: MTDE is hailed as the premier manufacturer of caving harnesses as well as other gear. The Club is their ultralight cave harness.
www.innermountainoutfitters.com
I am not affiliated with IMO, other than being a happy customer. The owner has young kids that he takes vertical caving & rapelling, so he would be a good resource regarding kids' harnesses.