Might want to get in touch with Will Piatt. It is probably going to be a Korean barrel or possibly Pyramid Air who is known to have some parts and barrels and you have to phone them because it may be too difficult and time consuming to navigate through their site for odds and ends parts.
Have you tried contacting Crosman? They do sell a .357 Benjamin Bulldog and would think they have barrels for them to sell as parts.
I can tell you that the barrels are accurate because I own a .357 Bulldog that shoots Nosler bullets, JSBs, Predator Polymags and Air Venturi slugs of various shapes and weights very accurately. I will say it is very much pellet and slug friendly compared to a Benjamin Marauder barrel.
CA
. I want something bigger.