John,
When I have a really badly fouled barrel of unknown history, I just put a chamber plug in and use a plastic oil additive syringe to fill the whole barrel with cleaner. After checking for leaks, how long I let it sit depends on how aggressive the cleaner is. Shooter's Choice is not very aggressive, so I'd probably leave it overnight. That works well with Butches Bore Shine, too. The maker told me he'd once left a piece of barrel steel soaking in it for six months and microscope examination showed no sign of etching in all that time. So it's good for a soak treatment, too. In the morning pour the stuff out into a jar to see whether it turned more blue or not? The soak will also help soften old carbon for patching out.
If your cleaned barrel (absolutely no more aquamarine color on the patches) fouls to the point accuracy actually decreases in fewer than 10 shots, you may have to do something even more extreme than JB, such as hand lapping or firelapping. Slugging the bore with a pure lead slug will reveal if it also has a constriction in that ares, which would be another vote for lapping. But try the mildest thing first, which is the JB.
Also, fouling tends to be greatest where the pressure is greatest because that's where pressure upsets the bullet hardest against the bore, creating the greatest friction. That's usually in the first couple or three inches after the throat. Unless your fouling pattern is different from that, I would concentrate the effort there. Use the patch method Jim Watson recommended but go back and forth a couple of times in those first few inches for each full pass down the bore you make.
