My experience, is that there is no way to avoid this with Barnes bullets. Since its an all copper bullet, after the bullet is carved by the rifleing, it still leaves a copper trail all the way out as its pure copper and does not have a jacket.
Comparitively, a lead bullet with just a copper jacket does not leave as much. As I heard it explained to me: as the rifeling carved into the bullet penetrates through the jacket to the lead... it leave less copper fouling than a Barnes which is copper all the way from the outside in.
I personally have found Barnes bullets to be AWESOME.. I love em, BUT, I also know when sighting in with them, that they are going to leave more fouling and cause a little more work to clean the barrel than another bullet.
I have NOT expereinced a tighter barrel. But with enough rounds, I would not doubt it.
To get out the copper use a solvent as others have said. Personally, I like Butches Bore Shine because it is not corrosive and I can leave it in the barrel all night to do its work. Even still, sometimes that does not do it entirely, and a brass brush in needed coated in solvent to loosen it up.
Best of luck!