Give what you said up in Post #14 about how you want to use the gun here's a few things to consider.
Any fixed sight gun is only going to shoot to POA with a select set of bullet to powder recipes. Basically each bullet weight will shoot accurately to POA with a given muzzle velocity. So your ammo options will tend to be limited. If you reload your own that's not a bad thing since you can tailor your loads to suit your needs.
A near full power heavy hitter at a given velocity can be your woods/hunting carry. A lighter bullet with appropriate muzzle velocity can be your plinker since that bullet will need a lot less powder to hit the correct speed to hit at the POA.
For example looking at my Lyman reloading book it seems like a 250gn hardcast with a nice wide meplate nose running at around 830 to 850fps will be around the maximum you want to push through a clone. A steady and frequent diet of such loads would likely see the gun tend to run into mechanical troubles fairly soon in it's life. But as a woods load and for occasional practice these would be just fine.
A more gun friendly "plinking" and everyday sort of load would then be a 230 gn hardcast pushed to some lower velocity that results in the bullet exiting at the same point in the recoil rise so that it hits at the same POA as the 250.
Going down to a 200 and the resulting powder to achieve the same POA would result in a truly poofter load that would be suitable for even the most squemishly recoil sensitive folks you can find.
The sights on the gun may or may not happen to regulate nicely to the 250 gn bullet shot at around 830 fps. Hopefully they hit a touch low so you can file off the top of the front a hair to raise it up to where it just matches the POA and POI.
Of course ALL of this can be avoided if you were to select a Ruger Blackhawk or Super Blackhawk. The adjustable sights remove all this nonsense. But then the adjustable sights guns don't have the smooth look of the old classic Peacemaker either. And that is certainly something to consider.
AND YET ANOTHER OPTION! ...... At the last moment I looked at the Uberti offerings. I see that they sell a Cattleman that is chambered for .44Mag. Now without getting into all the old .45Colt vs .44Mag things again it's worth noting that the extra metal in the walls of the cylinder found on the .44Mag allows for considerably more chamber pressure. It opens up the option of pushing a relatively heavy 250 to 255gn flat nose bullet at up closer to 1000fps without unduly punishing the gun. This would certainly provide more penetrating power. From there the same issues of POI vs POA for lighter bullets to create reduced plinking loads returns. I'd guess that a nice plinking/target shooting load would be something more like a 240 gn bullet with the powder reduced to achieve the same POI from the sights.
Now at around 1000fps you're still only part way up the range of what a .44Mag can achieve and still be within SAMMI specs. But out of deference to the action of the gun and to try to encourage a longer life from it before it shoots itself loose I suggest that this mid range power level would be more sensible than shooting full house loads.
The .44Mag isn't period correct of course. But there IS some precident in the simple fact that there was a .44 size round back in the day in the form of the .44-40. So if you selected a .44Mag version it would not be a case of total blasphemy...
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Then, of course, there's always the option of .357Mag. But I gather you're more interested in something that is more correct for the period. Or if you're not too worried then shooting full house .357's with 158gn bullets would be an option to consider. Or perhaps even jump up to the 180gn options. Then the 158gn bullets shot at around .38Spl +P velocities would be your regular plinking/target load.
