My question is whether you would use #3 or #2 shot if it were legal, and why? Also, why is the larger size legal in the first place? What's better about #2 shot for turkeys?
No, I would not use number 2 or 3 shot. It has been my experience that it's very difficult to get a decent pattern with even number 4 shot at 40 yards. Never patterned 2's at that distance but I'd be willing to bet that the pattern would be so sparse that it would be worthless for anything other than body shooting a turkey which I would never do. Yeah I'm sure number 2 pellets carry a lot of energy but that doesn't matter if you can't get one to land where you need it to.
Shoot 4's, 5's or 6's; pattern your gun; and aim for the head and neck.