The Judge is a handgun. A handgun with a smooth bore is classified as a handgun.
Negative
They even call it a pistol.
ATF Online - Firearms - Guides - Importation & Verification of Firearms, Ammunition and Implements of War - National Firearms Act Definitions - Any Other Weapon
26 U.S.C. § 5845(E)
(e) Any other weapon
The term ``any other weapon'' means any weapon or device capable of
being concealed on the person from which a shot can be discharged
through the energy of an explosive, a pistol or revolver having a barrel
with a smooth bore designed or redesigned to fire a fixed shotgun shell,
weapons with combination shotgun and rifle barrels 12 inches or more,
less than 18 inches in length, from which only a single discharge can be
made from either barrel without manual reloading, and shall include any
such weapon which may be readily restored to fire. Such term shall not
include a pistol or a revolver having a rifled bore, or rifled bores, or
weapons designed, made, or intended to be fired from the shoulder and
not capable of firing fixed ammunition.