I have no idea why you would want more than 2 oz. Thats PLENTY of shot in any of the turkey load flavors. A while back, we did some 3.5 inch, 1 3/4oz loads ( number 6 shot if memory serves correct), stoked to the ludicrous end of hot and got some pretty impressive chrono numbers and pretty good patterns. Again, this was in a loooong barrel with a hand made choke (with doggs), porting, back bore, forcing cone work, the whole nine yards.
When guys comne to have a gun built, the first thing I ask them is "how much work do YOU want to put into shooting this thing"? At some point, you start to loose the practical nature of the gun. Box stock guns shoot shelf bought ammo better than most shooters can shoot.
I like 2 oz loads, for sure as a stand alone go-to load for stock guns. Anything more than that, you are really pushing your luck as far as patterns go due to crowding of the bore, which causes flats on the pellets and as we know, balls fly better than boxcars. You need an over-bored gun and a very long choke to get the most out of the sledgehammer payloads.