There is one JHP round designed for .357 lever action rifles, the Hornady Lever Evolution series, which has a controlled expansion projectile that takes into account the much higher than (pistol) design velocities those get driven at out of the longer rifle barrel. Things like the regular soft points that may work well as pistol hunting rounds are alleged to disintegrate in the target and come apart very fast out of the rifle barrels and fail to penetrate. My experience with the .357 in lever rifles is seeing the factory 125's, HP or solid, driven at over 2000 fps. Way too fast for the bullet design. The other alternative is hard cast bullets for hunting which will certainly perform but I would personally stick with the gas check loadings, again because of velocity.
I owned 4 Uberti's during my CAS competition days, including one identical to what you ordered, and they all performed great, but the design is the old Winchester toggle link, which is much smoother yet weaker than the later Winchester 92 action. The stronger steels we have today make up for a lot of that and make them viable smokeless cartridge shooters, but if I were going to see just how hot I could load a 357 round, I would prefer to try that with a Win'92 or Marlin action. I would stick with the factory spec ammo only and not hot rod my .357 handloads.
If I loaded up 50 rounds of my intended ammo and ran it hard (ran the action vigorously) I would consider the rifle fully vetted. If you are looking for anything more than that you are making too much of a big deal about it. Again this is not some semi-auto which is in any way going to be ammo or chamber sensitive, nothing to vet on those counts. The rifle runs everything from the mildest .38 special loads to the hottest factory .357 loads and simply does not care other than the differences in group sizes you get with various ammo. Timing problems with the lifter were the most common issues I saw in mine and many other '73 actions in all my years in CAS. It is a fiddly action and usually when people start to go for the aftermarket short stroke kits is when those problems start to appear. There are aftermarket lifters in some of those kits that are bent to install them and establish timing but guess what, if you bent it to install it, it might get bent front running hard. The screws Ubertis use are soft and that is annoying, the lifter springs are fiddly and there are aftermarket versions that last longer but as long as you keep those screws tight they are ok.
In SASS I completed with a 125gr .38 special cartridge that was doing 800 fps tops from the rifle and ran them as fast as I physically could and they did great. In NCOWS competition I ran them with a fully packed 2Fg black powder .357 case behind a 158gr and it ran great, extremely accurate. Out here on the farm I ran them with the Hornady Lever Evolution 140gr bullet in a .357 case that clocked about 1650 fps and woe to the coyote who got in front of my sights. It is a versatile rifle.
They are simply not compatible with scopes because there is no good place to mount one on the receiver, both from mounting and ejection considerations. Somebody could forward mount a red dot or LER on the barrel but please don't show me the pictures of such a travesty. I might cry.
Arguably the Japanese made Winchesters are more precisely fitted and better built, I never owned one but @
Malamute has one I know. He can chime in on more things he knows about the basic desgin.