I don't have that particular lathe, but by the looks of it, it seems to have a fairly standard chuck mounting system. Adding a 4 jaw would seem to be a matter of picking a 4 jaw chuck, machining a backing plate and assemble.
Not surprising that BB service is of little help. I've got a CX601 BB mill and their customer service is horrible. With BB, you're pretty much on your own once the lift-gate truck drops it in your driveway. They even do everything they can to get out of a valid warranty claim. Been there, done that, got screwed.
I won't even buy a pencil off them anymore. They charge you more for less and then turn their backs on you.....stuff they sell is all the same as ebay/china stuff anyways. The only difference is they charge you a premium for buying from a "brick and mortar" store.
You'd be far better off to pick a quality, aftermarket 4 jaw chuck and make/buy a backing plate than buying BB's overpriced "chinesium". BB sucks....