If I had to choose I would go with White's Treefrogs(Dumpy Treefrogs) or Firebelly Toads, or a Toad like an American Toad. I think you even have Woodhouse's toads in your area, which I keep. I love those guys! White's will require good vertical space, and Firebellies Horizontal space with about 60% water to 40% land. Toads just require a terrestrial setup with a water dish, some sort of substrate like coco fiber or topsoil. It was really good of you to explain frogs aren't for handling, but I've seen plenty of people handle White's Treefrogs, but that mostly consists of letting them perch on your hand. Firebellies produce a mild irritant though, so that may make them undesirable. If he doesn't want toads, then I apologize, but they are just land dwelling frogs lol. A lot of people recommend Pacman frogs as a good first time frog, but they can bit and have small "teeth" called odontoid projections, and I feel that might make them undesirable given your son's needs and/or desires. I find my Woodhouse's aren't particularly stressed by handling. Pretty fair sized at around 4 inches full size, so if he likes toads they are a good option. The worst they've ever done to me is tried to pee on me.
LLLReptiles is a good supplier of frogs, feeders and supplies, and Josh's frogs carries Dart Frogs, a couple treefrogs and lots of frog supplies.
I hope this helps. I'll add more later.