Hi Black Jaque;
I have two suggestions for you.
First, if you do want a standard production rifle. Thompson Center made a model called the White Mountain Carbine a number of years ago. It had a 12 3/4" trigger pull, and an overall length of 37". It weighs a little over 6 pounds. It does have a factory installed rubber recoil pad, which although not period correct, is an excellent way to avoid having the recoil beat up on a youngster! An ideal kid's starter rifle.
You could find one on the used market. Also, I have one, but it's promised to my nephew's son if he wants it. I'll be talking with the lad October 23, and if he doesn't want it, I'd be willing to sell it to you. Let me know if you're interested.
Second idea is this. Pecatonica River Muzzleloader Supply makes a "Beck Boy's rifle". It is a deliberately downsized rifle, patterned after the original Beck rifles, with young fellows such as your son specifically in mind. It's exactly what you're looking for. They downsized it specifically to keep everything in the proper relationships, so you won't have to butcher it up and still have a cheekpiece or something out of proportion.
You should call Dick Greensides, owner of Pecatonica and talk with him. Pecatonica is a first class "kit" supplier, and I can personally attest to the excellence of everything they offer.
Pecatonica's phone number is 815-968-1995. They also have a modest website, but it doesn't do justice to the quality of their work.
Regards,
Jerry