For my nephews, I took a Crosman (Mendoza) Quest 800, which I had shot enough that the spring was fatigued and it was probably at that point a Quest '500' (fps that is), then removed recoil pad, sawed about 1.25" off the stock, and re-installed recoil pad; voila.
But for a 5 year old, a lighter and cheaper and more versatile one would be the old standbys of pneumatic multi-pumpers like the old Daisy Powerline, Crosmans, etc. They are light, shoot either BBs or pellets, and you can pump them just 2 or 3 times to get a little no recoil puff shot out of them. Used to be about $40.
Only problem with those is, if the LOP is too long, they're plastic, not wood, so re-attaching the buttplace after the saw job could be a problem. Probably nothing that some superglue won't solve.
These do have rifled barrels, so personally I wouldn't shoot BBs from them.