Barrel cleaning method
I don't use corrosive ammo in modern firearms, however, I used to shoot black powder and the residue from firing which while not having the salts problem, is also damaging to barrels, and anything else metal.
The method I used to clean the barrel was to get a bucket of HOT soapy, and a bucket of plain hot water each about half full. Remove the barrel from the stock remove the nipple, place the barrel nipple end in the water soak enough patches to make a snug fit in the barrel push the cleaning rod down to the bottom of the barrel and pump the rod up and down . The patches make a tight enough seal to cause the water to be pulled into the barrel and pushed out through the nipple opening. Switch to clean water, after a while the barrel will become warm to the touch and will quickly air dry carefully dry inside and out, and oil. this method could easily be adapted to any firearm barrel, but requires some disassembly, but you already have all the stuff you need.