Sounds like I'm alone here, but I have two sons that are now 16 and 18 and neither owned their own youth gun. During bird/rabbit season they used a 20 gauge 1100 semi and a 20 gauge pump or single shot. They used those from the time they could go with me (around 9 or 10) until now. Knowing they shot smooth I never had to worry about scaring them with something lighter that may kick too bad.
For deer, they always used a muzzy because they shoot so smooth with zero kick. By the age of 12, they also used a 12 gauge 1100 for deer.
If you have a gun that your grandson can handle well enough, I'd let him use that and someday pass it down to him. Just my opinion. If he shows enough interest that he will actually walk around alot hunting bunnies or birds, then maybe I'd get him something he can call his own that is lighter to carry.
Nothing will drive more crazy than if you buy him a gun and then he never wants to get out and use it. Have you tested the waters with him yet?