Without knowing the legalities vis a vis turkeys where you live, I'd actually suggest a light RIFLE for this. A scoped semi -auto .223 of your favorite flavor and body shots - flat shooting, certainly lethal. Not sporting, maybe some of the birds wouldn't even be edible... but this is simple population control. The ones you shoot will never come back, for sure. Shoot as many as you can, but aim for the hens first.
If it HAS to be a shotgun, a highcap semi-auto with a tight choke and lots of #4 Buck. Again, body shots. On the smoker might be what you WANT them to be, but DEAD is all you NEED them to be.