If you need a Lewis Lead Remover to clean your bore after each range session, then I think you have a problem with your lead bullets. The LLR should be reserved for major lead fouling problems. A good lead bullet load should not require its use.
I am not familiar with IOSSO bore paste, but, if it is anything like other bore pastes, it will be slighly enlarging your bore and smoothing your rifling. A little of that done PROPERLY can be a good thing. BUT, if it is done in a manner that wears the bore or cylinder throats unevenly or out of alignment, then it is eventually going to damage the accuracy of your gun. That would be a shame, because your gun seems to shoot the 125 gr Remington factory loads very accurately right now.
Based on what you have told us so far, I think your probem is most likely due to using those lead bullets in less than optimal conditions. But, there are a lot of issues with lead bullets.
Before starting on those issues, it would be best to make sure that your die set-up and other loading parameters are working to avoid accuracy problems (e.g., bullets started straight so that they are not cocked in the cases when seated, case same length and mouths square so that crimps are uniform and not lopsided, etc.) That can be done with the jacketed bullets that you just loaded and any powder reasonable for the cartridge. But, you will have to do a load work-up with the powder to see what charge weight is the most accurate. Just picking one or two charge weights to test can easily miss the weight range that provides good accuracy in YOUR gun.
Another thought to consider: why are you going to 158 grain bullets? Most manuals seem to talk about using 180 grain bullets in the .357 Magnum when they are working up silhouette loads, especially for the rams. If it is more power for topling rams that you are seeking, then a heavy bullet moving as fast as possible seems like the goal. You can usually push hard-cast lead bullets faster than jacketed bullets of the same weight, but that isn't a good idea unless you can make them just as accurate as the jacketed bullets. That means good bulllets, good loading technique, and careful charge weight work-up for maximum accuracy.