car haulers, moving vans, flat beds, dry vans, refers... there are big bunks pulling all kinds of freight.
You won't likely see big bunks pulling tankers- bulk or otherwise.
A big bunk- depending on the setup- can weigh anywhere from 24,000 on the low end to upwards of 30,000. That's leaving out 90 inch sleepers which are just a step up from a regular bunk. I don't call it a big bunk unless it has at least 120 inches of sleeper. 30,000 lbs is a massively heavy big bunk- you're most likely to find them weighing 26,000-28,000. That's been my experience from talking to other big bunk owners.
I've recently seen big bunks with Fedex (non-carhaul and carhaul), Landstar, Air Ride and many other traditionally non-big bunk companies.
Most of the large sleepers I have seen are usually pulling a flatbed.
Either Landstar or Mercer. You might want to start making some calls to different companies and get your info straight from the horses mouth though. That's the best way to find out for sure if they will accept one.
I think they are called FAF. Foward Air. I know they haul all light frieght in dry vans and they are all O/O fleet and I have seen some big sleepers running for them.