So, the simple answer to the question is yes, you do have to have the paperwork in the cab with the driver, but not because it's HAZMAT, but rather because
FMCSA regulations state,
"A copy of the bill of lading must accompany a shipment at all times while in your (or your agent's) possession. Before the vehicle leaves the residence of origin, the bill of lading must be in the possession of the driver responsible for the shipment." This is a requirement for all shipments, HAZMAT or not. But you do not have to leave it on the seat if you exit the vehicle, or have that paperwork tabbed, or any of the other special provisions that must be made for HAZMAT paperwork.
Well, I guess a debate about the right answer will be necessary. Turtle is incorrect. See: 49 CFR 177.817 Shipping papers.
A HAZMAT shipping paper and a bill of lading are not the same thing. If the material in question triggers the requirement for a HAZMAT shipping paper, the shipping paper triggers the following, and it does not matter what kind of vehicle is being driven. Vehicle type does not matter, quantity does not matter, the requirement to have a HAZMAT-endorsed CDL does not matter.
The question is about whether or not HAZMAT shipping papers must be carried in a particular way. The answer is yes. The shipping paper triggers the requirement. If a HAZMAT shipping paper exists, it must be treated as follows:
(e)
Shipping paper accessibility—accident or inspection. A driver of a motor vehicle containing hazardous material, and each carrier using such a vehicle, shall ensure that the shipping paper required by this section is readily available to, and recognizable by, authorities in the event of accident or inspection. Specifically, the driver and the carrier shall:
(1) Clearly distinguish the shipping paper, if it is carried with other shipping papers or other papers of any kind, by either distinctively tabbing it or by having it appear first; and
(2) Store the shipping paper as follows:
(i) When the driver is at the vehicle's controls, the shipping paper shall be: (A) Within his immediate reach while he is restrained by the lap belt; and (B) either readily visible to a person entering the driver's compartment or in a holder which is mounted to the inside of the door on the driver's side of the vehicle.
(ii) When the driver is not at the vehicle's controls, the shipping paper shall be:
(A) In a holder which is mounted to the inside of the door on the driver's side of the vehicle; or (B) on the driver's seat in the vehicle.