I winced when I heard about dragging a copper patch through the barrel.... stainless is definitely stronger than copper, but I wouldn't do it. I get stubborn fouling out using a "next size up" (cleaning a .38, use a .40 brush) NYLON bore brush dipped in automotive chrome polish. Must be nylon. Chrome polish is very mild in abrasive, so you can really scrub at it without worry. To get the garbage near the forcing cone end: put some polish on a Q-tip and work it in. Then take the brush off the rod and work it in and out by hand at the cone end. Run patches on a jag and it will be clean. If anything is still in there after that, J+B bore cleaner on a patch will get it. That has some mild abrasives so only use it for really stubborn stuff (like lead).
BTW: chrome polish is the best for getting the burn rings out of a wheel guns cylinder tubes. I put the rod in my hand drill, dip the brush in the polish, and spin it in the tube. About ten seconds usually does it.