Been there, done that...
The ring is a bugger to get out of the tub.
Next time try a bucket of soapy water with a shotgun swab on your cleaning rod.
The wet swab creates a suction in the barrel that pulls the water in and out, or you
can just pour the water into the barrel.
I drew the water out of the tub; I didn't use the tub for cleaning . Then I used
the method you described. I also started the tea kettle to get hot water to pour
water down the bore when I was finished scrubbing. Finally I used a patch coated
with gun oil to coat the bore & outside of the barrel (I also removed the lock
mechanism and cleaned the underside). The nipple was placed in warm soapy
water while I cleaned the lock & barrel ; I cleaned and dried the nipple then used
anti-seize compound on the threads as I reinstalled.
Blessings,
Longbeard