Lead cutter loads
I've never seen lead fouling in any of my revolvers (which see much use and few jackets) that could not be cleaned out in 50 strokes of a proper brush (slightly larger than the hole being brushed). I do prefer to eliminate or at least drastically reduce even that small amount of labor by using lead cutter loads.
These are hand loads, and for .357 I use 3.5 gr HP-38, Lyman 358356 in wheel weight, remaining case space filled with Cream of Wheat. In .45Colt I use 6 gr HP-38, Lyman 454424, and Cream of Wheat.
Six shots usually does the trick, once I used 12 to clean up after .45Colt loads - Lyman 454190 in wheel weight at 1300 fps.
If you do use a brush, use rifle brushes. They're longer and do more brushing per stroke. Use bronze, brass, copper - anything harder than lead and soft than gun parts.