Have you thought about getting one of those yildiz single shots and cutting the stock down, then adding a recoil pad?
I've thought about it. I borrowed a NEF .410 "youth" single shot thinking it would be small/light enough but it's not. It weighs a surprising 7 pounds and has a 22 inch barrel. I've thought about trying to trade for the NEF and cutting it down some.
We're wanting to try busting some clays over Thanksgiving; I'm not sure he's big enough (only 49 pounds) to swing a shotgun, but we're willing to try. Even a 5 pound gun would be 10% of his bodyweight so it might be a bridge too far right now.
He did shoot an appleseed event last weekend with a short barreled .22 rifle I did the paperwork on & made for him and did very well, "passing" their qualification test (harder than you'd think). *Although I kept him in prone for all stages. He's got that .22 rifle that fits him and we've been shooting the Diamond Atomic Bow which is a great bow for kids BTW; once I get this shotgun thing figured out he'll have the complete set.