I use hot soapy water followed by flushing with clean hot water. The cheapest way and it's worked great for me for 50 years.
I've studied the Civill War for a long time and have never run across a first person account of urinating in the barrel. Could you suppy us your source for that?
In a privately published book on the 1851 Navy that is a collection of information gathered by a gentleman from Civil War books, he states the practice of cleaning revolvers - hot water with lye soap added in a container where the revolvers were hung down in with the grip fram resting on the rim. I've run across this several times over the years. It makes sense in the field an then flushing with clear water and drying.
If you've ever been over on the shotgun range at Friendship,you'll see tubs of water where the barrels can be removed from the stock after shooting rounds and the breech can be submerged to clean the barrels by using a cleaning rod to pull the water up in a pumping motion. Works good on the shotgun barrels. When I'm down there, I usually just boil some water on the fire and use that - dry and then oil.
Some folks use date/windex mixture. Everybody has their own method. The important thing is to get the gun clean, dry and then a coating of something to prevent rust. Personally, I'm not in to all the fancy solvents and cleaners. Soap and water was pretty much used by our ancestors and it worked well for them.
If a pair of '51 Navies were good enough for Billy Hickok, then a single Navy on my right hip is good enough for me . . . besides . . . I'm probably only half as good as he was anyways. Hiram's Rangers Badge #63