The suggestion to soak with Kroil is good, but I think you'll find plugging the bore with a stopper and filling it with something that is really capable of softening hard carbon will work best. I like Gunzilla
for this. It is vegetable-based and not essentially non-toxic and in about 24 hours will turn the carbon to soft sludge or tar that brushes out very easily. It has the advantage that it also does a really excellent job of loosening rust so any trapped in the bore will brush out with it.
My second choice for this same job is Ed's Red. It sure costs a lot less, and since it is home made, is cheap to do. You can download a PDF file of the recipe from my file collection, here
. Unlike penetrating oil, it has both polar and non-polar solvents, so it tends to get all forms of fouling except metal fouling. I would recommend plugging the bore and letting Ed's Red sit for several days to a week. You'll probably find the carbon all sitting in a sludge puddle on the stopper when you pull it.
The second thing that can clog a bore is metal fouling layers alternating with carbon layers. I suggest, after the carbon removing step, you want to squirt a little brake cleaner or Gun Scrubber down the bore to rid yourself of the cleaner from the first step (it's not a good idea to mix cleaners; Gun Scrubber is just solvent that evaporates off) and re-plug and fill the bore with a copper removing cleaner. I like Boretech Eliminator and their Cu++ products because the are very fast and essentially odorless, but they're not cheap. And if you are going to let it sit, any brand copper remover will work eventually. An Outer's Foul Out is good for this kind of work as long as no rust of any kind was present. They warn you that it can etch a barrel if it started with rust in it.
When you pull the stopper on the copper remover, you will see blue from the copper reaction if it was present. The only exception is with KG-12, which only turns tan.