For your purposes, a 20 gauge would suffice with lead. They will hold a good enough pattern at 20 yards for turkey, and even 25 yards with the right combo. For big game, I'm going to assume you would be using a round ball. A 20 is definitely enough for upland game and small game. With non-toxic shot, that's going to depend. I'm not sure bismuth from a 20 is going to be very effective on turkeys. I'd spring for one of the heavier shots, Nice Shot #7.5 if you can get it, or even #9 TSS. Bismuth will work fine for small game. I have no idea about casting round balls of non-toxic stuff.
Don't discount the larger bores either. I'm a big bore fan, through and through. If I could duck hunt with an 8 gauge, I would. I'm a big time 10 gauge fan. 12 gauge/75 caliber is as common as it is for a reason. The big bores can still be loaded down, and they can also throw a lot of shot if you want. I wouldn't shoot steel from a muzzleloader. There's nothing good about steel shot, it is a raw deal all around. I think Nuthatch is exaggerating a little bit, but there's no doubt that steel shot from a muzzleloader is not very effective. Velocity is only part of the factor. You could step up in shot size, you would be looking at 1 or 2 sizes bigger than a modern shotgun shell. For ducks that's probably going to be #1. To get a pellet count to even remotely be effective on ducks, you'd be looking at shooting 1 1/2 oz or more, which is about the same as a modern 20 gauge shell with 1 ounce of #3 steel, but your muzzleloader probably wont pattern as tight, making the problem even worse. That's not something I'd shoot from a 20 gauge, it's a load for a 10 gauge. Bismuth is a big step up. #4 has the penetration for ducks to at least 40 yards, and has a fair pellet count. For small/upland game, I'd try #5, and maybe even #6. Bismuth is about like lead plus 1 size, so #4 bismuth, is like lead #5. The only other shot I'm aware of that can be fired with no protection is Nice Shot, which is an equivalent to lead... Except at $40 per pound. Anything else, ITX, Hevishot, HW13, TSS, etc. needs a modern thick plastic "steel shot" wad to protect the bore. I've fired plastic wads myself, and have not found them to be the problem many have found them to be. I've not shot 50 in a day though, but 5-10 is no issue.
I'd consider waiting a few months (hopefully), and Jim Kibler is coming out with a trade gun that will fit this description. For non-traditional single barrel guns, there is a TC new Englander in 12 gauge, but they are a bit pricey for what they are. If you can get one under $400, jump on it, but it probably wont happen. For the price people are asking for them, you may as well buy a custom (Kibler), or simply buy an actual original fowler or SXS. You will probably want a single barrel gun if you want to hunt big game with this. A SXS is fantastic for wing shooting, and turkey hunting, but it's a lot of variables in POI for shooting a single ball.