Here's an idea. Try a Black Powder cleaner. I use Lyman's (came with the guns), and it does a great job on BP carbon fouling. Shooters choice and Hoppes each make one, I think.
If you have ever shot BP, then you know how heavy the powder/carbon fouling can be, and how nasty it can be. I have been very impressed how easy the clean up of my BP guns is with the Lyman's - they are easier to clean with it than my centerfires are.
Also try Breakfree's Powder Blast and CLP. Something that I haven't tried in my shotguns (12 gauges can foul with carbon badly) is WD-40, and that seems to be popular among the trap and skeet set. You might consider it. Powder Blast and CLP are the only ones that I have had direct experience with.
As for soaking, I have done so with the Breakfree products and it should be alright with WD-40, but have never tried it with the Lyman BP cleaner. Check the labels first (because of rusting), or even call the manufacturers first to find out if it is o.k.
Good Luck.