Howdy Again
The most damage that happens most often when cleaning any firearm is the cleaning rod rubbing against the bore near the muzzle. It is impossible to clean most revolver barrels from the cylinder end with a rod, so if you want to do the least harm, use a Bore Snake and clean the bore from the cylinder end. Same with the chambers, you can clean with a Bore Snake instead of a cleaning rod. But I will be frank with you, I have found over the years that I can remove more crud with less elbow grease with a cleaning rod than with a Bore Snake. The Bore Snake only has bronze bristles at the end, and it has to be inserted and run through the bore many times to get the same amount of cleaning as running a rod with a brush on the end a few times, because I can run the rod back and forth lots of times without removing it from the bore. Frankly, if you are only going to shoot it a few times, and clean it afterwards with a brush on the end of a rod, you are going to have to try pretty hard to do any damage to the rifling near the muzzle. You can buy aluminum rods that are softer than the steel, you can also buy steel rods that have a plastic coating on them so they don't abrade the steel of the barrel.
I don't worry at all about finding 'clean burning' ammunition. As I said, I shoot some of my antique cartridge revolvers with Black Powder all the time, and Black Powder is anything but 'clean burning' I just have to remember to clean them afterwards. For modern Smokeless ammo, forget about trying to find 'clean burning' ammo. I can't tell you how many times I have heard guys on the firing line at a CAS match complaining because their ammo is not 'clean burning'. Their concern is that a little bit of Unique leaves some burnt powder flakes behind and they are concerned this will affect how well the guns function. We don't shoot fussy semi-auto firearms in CAS, a little bit of burnt powder in the barrel is not going to cause them to jam. Forget about trying to find 'Cowboy Action Loads'. There is nothing about them that is necessarily any cleaner burning than any other Smokeless ammo on the shelf. Trust me on this. Particularly these days when there is an ammo shortage because so many guys are panic buying guns and ammo. Just buy what you can find, shoot as much or as little as you want, and clean carefully afterwards.