YOu should consider buying a utility grade stock to use for the kids gun, so that the cuts you make won't destroy an other wise good, factory stock. If the factory stock is birch, well, make your cuts on the factory stock. When the child gets bigger, and can handle a full factory stock, you can always buy a stock made by one of the companies, using a better grade of wood, to put on the gun.
I have seen several young kids shooting shotguns their fathers cut back for them, and then added 1/2" pieces as they grew up. They were not necessarily pretty, but they work quite well. And you can alter the pitch of the stock as the child grows, to take into account the fullness of the face, and a barrel rib cage, common to women, and to some boys.