If you are insistant on shooting the Beretta for turkeys just go buy some quality 2 3/4" shells in #6 shot like the Federal Pheasant Forever shells at 1,350 fps. Pattern them using a turkey head target and see what load/choke give you the best pattern. You don't shoot that many shells while...
I'm not following your issue with the 16 ga.
A hunt at Granite Falls would be good. I've known Terry and his dad, Richard for years. We had several NSTRA field trials there.
In the article I read several years ago it said a pheasant could have a crop full of soybeans and literally starve to death due to their inability to digest them.
Jon, there are new A5 20's on GunBroker. I look yesterday. Remember the 20's (just like the older Auto 5's) are built on the 16 ga. frame. They are only a couple onces lighter than the SweetSixteens which I suppose is due to the smaller barrel.
It's actually a Remington 105 CTi. They had cycling problems that Remington was never able to solve. Here's a good article about them.
https://www.outdoorlife.com/guns/remington-105-cti/