Here are a few thoughts.
Ability to use the same ammo doesn't matter. I am not saying this to argue but, really. It only matters on the firing range or when you are hunting with others who don't know what they are doing. If the ammo is good for short range in a handgun, it won't be good for long range in a rifle, and vice versa. You will be packing an extra 6-7lbs of gun to get 300fps, or an extra 2.5-3lbs to lose 300fps. If you are planning on carrying two types of ammo for the two different guns, then it doesn't really matter if they take the same load.
.22 is chambered in lighter guns, the ammo is smaller and lighter, cheaper to buy, and more likely to be practiced with. It is easier to carry 200rds of .22 than 50 .357, and often the gun will weigh less as well. While the .44 special could make an excellent small game load, it takes quite a few .22LR to add up to the same amount of weight.
If the guns could be "X" caliber and .22, then you would still only need one set of dies, brass, etc. Also, you wouldn't be limited to revo cartridges for your rifle. Then two arms could be used that could each fill a specific purpose well, IE. small game, long range larger game.
At that point, it would be a question of duration. How many rounds of .22/small game ammo would be required? If less than a couple hundred, then a specialized load for the larger arm may weigh less then a small game handgun and equivalent round count.
I don't mean any of this as an argument, just food for thought. As far as the best dual arm/single ammo combo, I think it would depend on how many rounds you needed to bring vs. how large an animal you thought it necessary to go after.