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To bring this back:
I do have a 12 Gauge Beretta Pintail. I have a Jelly Head choke and 3″ Federals #4 that I have shot decent with. I have not tried to pattern that many different shot loads out of this gun. Any suggestions. 
Rob, is that gun back-bored like the Brownings? I’ve found that these guns can be a different beast, but in a good way. They’re generally less picky about ammo, and your patterns tend to do at least decent with most choke/load combos.
That said, I’m a big fan of #6 shot in the Hevi-variety. Whether it’s the Federal Heavyweights, Win. Xtended, Hevi13, etc, I think the overall goal is to put as many pellets in a small area that’s the turkey’s head, as possible. This is more easily achieved in most cases with more pellets that you get by switching to smaller shot sizes. Previously, with lead, all this did was create a load that would pattern well to 50, but only kill to 40.
However, the hi-density shot nowadays hits alot harder, almost like one size up. 6’s hit like 5’s in other words, and I’ve killed a few birds flirting with 50+ yards. Not that I encourage that type of shot, but if you know your gun and its limitations, why not?
In my Winchester 1300, Remington 870, Stoeger M-2000, and Browning Gold, the old Rem. Hevi shot did well, but Winchester X-tended has outperformed everything else that’s been put up against it. I’ve had the worst luck with Federals overall.
I’d run Hevi 13 and Winchester Xtended in #6’s through it and see how it patterns. Not sure what constriction your choke tube is, but it would be nice if you could borrow a buddies for comparison!
Joel
LIP 
You little guys 