..I don't know how the accuracy is on the rifled Mossbergs.. but, if it was my decision, I'd probably go with a whole new shotgun, for about the same amount of money. You said money wasn't an issue, so, from what I've heard, the Hastings barrels are supposed to be premium barrels... I would probably get one of those for my Remington. I think you would be assured of an accurate shotgun.
My shotgun is a Mossberg, and I'm about to get another in 20ga for my daughter, so it should be evident I like them... but I just don't know about the accuracy of their rifled barrels. If I knew they were adequately accurate, I'd buy the whole new gun in a heartbeat, and not have to worry about taking the guns apart everytime I wanted to shoot clay pigeons, or deer.

"Progress, far from consisting in change, depends on retentiveness..those who cannot remember the past are condemned to repeat it." Santayana