lead
I have not shot a gobbler with a 20 ga. But I started my turkey addiction with a very old vintage 12 ga pump (grandad's Savage of 1921) and shot 1-1/4 oz of lead #6 from the old gun and it did fine until I retired it. That payload is the same you will get with a 3" 20 ga shell. Any of the lead loads you mention should give you decent results on gobblers as long as you test, and are careful with your distances afield.
As you discuss a modified choke, you patterns will likely be a bit more open than a full choke gun, all things equal. Therefore, I would lean to #6 lead as my starting point and pattern and test accordingly. There will be more pellets per ounce with #6, and theoretically, that will yield denser patterns. HOwever......I am a big fan of #5 shot,( heavier than #6, denser patterns than #4) and would shoot a few of those just for grins. I believe there is some merit to the slightly heavier #5 pellet being able to penetrate turkey bones better past 25-30 yds, , (spine and punkin) which is what you want to anchor a gobbler. For the same reason, I would argue against steel. Steel is lighter than lead of course for a given volume, and the lighter pellet will thus run out of energy as velocity drops. The whole "space shot" industry revolves around shotgunners being largely dissatisfied with steel.
I would be skeptical of #4 shot, as in my 12 ga guns, I have not seen patterns that seem tight enough to suit. That was especially true with grandad's old Savage. Likewise, head shooting gobblers is different than busting waterfowl, and #2 shot is just another step in the wrong direction yielding still thinner patterns. I have killed some gobblers with a swarm of smaller #7-1/2 shot, but that was under a set of circumstances where I could guarantee (decoys and blind, small open field) that the ranges would be short, (under 30 yds) velocities high, and thus penetration was adequate.