Buy a box of Stingers.
Fire a few. It doesn't have to be in the Buckmark since all you're after is the spent cases.
When cleaning the Buckmark, take one of the spent Stinger cases and push it into the chamber.
It will cut the "crud ring" in the chamber that's built up by the lube on the standard length, 40 grain, .22lr ammo.
It's the same idea behind using a spent .357mag case in a .357 mag that's been fired w/.38spl to remove the "crud ring".
I keep a few loose spent Stinger cases in my range bag. I shove one in the chamber to ream it out after every couple/few hundred shots with every .22 I shoot.
When you say WWB, would that be the Winchester Wildcats?
IMHO - Wildcats are lower on the food chain than Remington Thunderbolts.
Both are filthy, but, at least the Thunderbolts shoot straight compared to the Wildcats.