If you have a true carbon cake you need the JB or some kind of abrasive, just a tight patch, you'll be done in a minute or two. Just don't go after it like lapping, a few short strokes in the offending area and push the patch through then clean and inspect. Brushes are pretty much obsolete, way to much effort which translates into running that rod through critical areas more.
I'm a firm believer that changing cleaners does good, I've followed guys who cleaned to white patches with Boretech and I could see the carbon with my eye and the borescope revealed a mountain of carbon, I used several different cleaners and I'd get a little more out but eventually they showed a clean patch. One patch with abrasive and it came out black and gummed up, next three solvent patches were caked with carbon sludge, you just have to break the glaze or what ever happens and it'll come clean.
Just be aware that the abrasive patch will always come out black with either carbon or metal so just do a patch or two and clean normal no need to lap it out