It looks like all the confusion regarding whether the truck was commercial, had airbrakes, was registered Historic, etc has been removed from the 2011 edition of the California Vehicle Code (CVC) and you need a Class B license to drive it in Californis.
The M35A2’s unladen weight is over 13,000 pounds and it has three axles.
According to subparagraph (B) under the Class B License requirements (Section 12804.9 appended below) an M35A2 or M35A3 requires a regular (not a non-commercial Class B license with house car endorsement) Class B or Class A license to drive it in California.
It seems like the license requirements for the 2011 California Vehicle Code breaks down as follows:
Class A license:
Vehicles listed as Class A and all vehicles under class B and class C.
Class B license:
A single vehicle with three or more axles weighing more than 6,000 pounds; or
A house car over 40 feet in length; or
Vehicles listed as Class B and all vehicles covered under class C.
Class B license (non-commercial with house car endorsement):
Any house car over 40 feet in length.
Class C license:
Your regular car license in California.
Excerpts from the 2011 California Vehicle Code.
Section 12804.9 Examination and Driving Test Classifications
Examination and Driving Test: Classifications.
(1) Class A includes the following:
(A) A combination of vehicles, if a vehicle being towed has a gross
vehicle weight rating of more than 10,000 pounds.
(B) A vehicle towing more than one vehicle.
(C) A trailer bus.
(D) The operation of all vehicles under class B and class C.
(2) Class B includes the following:
(B) A single vehicle with three or more axles, except any three-axle
vehicle weighing less than 6,000 pounds.
(F) A house car over 40 feet in length, excluding safety devices and safety
bumpers.
(G) The operation of all vehicles covered under class C.
(3) Class C includes the following:
(C) A house car of 40 feet in length or less.
(D) A three-axle vehicle weighing 6,000 pounds gross or less.
Section 12804.10 Noncommercial Class B License with House Car Endorsement
(a) Notwithstanding any other provision of law, a person issued a class C license under paragraph (3) of subdivision (b) of Section 12804.9 may drive any house car of 40 feet in length or less without obtaining a noncommercial class B driver's license with house car endorsement as described in subdivision (b).
(b) Any person seeking to drive any house car over 40 feet in length, excluding safety devices and safety bumpers, shall obtain a noncommercial class B driver's license with house car endorsement as described in this subdivision.