I can only speak for California, which probably won't help you too much, but here goes anyway!
CA requires at least a class B commercial driver's license for the 2.5-ton 6x6 trucks. State law calls them "class B trucks" because they have more than two axles and are over some weight limit (I always forget whether it's 6000 or 9000 pounds). A bobbed deuce with two axles might be OK with a regular class C license, though it may be arguable whether an air brake certification would still be needed.
I get the impression from discussions here that CA's requirements are unusual, and that most other states wouldn't require a CDL for the deuce. The deuce falls under the weight limit for federal class B license requirements, but some states (like CA) are more strict.