Okay, you called me out.
Per the BATF office in Seattle. They directed me to 18 USC Section 922 (b)1 which states:
"(b) It shall be unlawful for any licensed importer, licensed manufacturer, licensed dealer, or licensed collector to sell or deliver --
"(1) any firearm or ammunition to any individual who the licensee knows or has reasonable cause to believe is less than eighteen years of age, and, if the firearm, or ammunition is other than a shotgun or rifle, or ammunition for a shotgun or rifle, to any individual who the licensee knows or has reasonable cause to believe is less than twenty-one years of age"
The inspector then directed me to the definition of a shotgun which is:
"(5) The term "shotgun" means a weapon designed or redesigned, made or remade, and intended to be fired from the shoulder and designed or redesigned and made or remade to use the energy of the explosive in a fixed shotgun shell to fire through a smooth bore either a number of ball shot or a single projectile for each single pull of the trigger."
When I asked what she would classify the weapon as, she said that it is, technically, 'unclassified' but clarified that it does not fall under the definition of an "Any Other Weapon" or "Firearm" as the BATF uses these definitions. These definitions are drawn from 1934 and 1968 laws and various other laws and rulings. Since these definitions are a matter of law, the BATF cannot create another category for "Pistol Grip Shotgun." They are also limited by the law in that they cannot classify it in any other category. As such, it falls under the standard rules for firearms which still means you have to be 21 or older.
There is a misconception that the law says, "You can't buy a handgun if you are under 21 years of age." In fact, the law states as quoted above that, "If you are over 17 and under 20 years of age, you can ONLY buy a shotgun or rifle by the definitions of the GCA of 1968." The categories for an 18-21 year old are therefore limited by specific inclusion rather than exclusion. Only shotguns or rifles.
I'd do a victory dance for having found this, but I am strangely saddened that these things are so unconstitutionally vague.
Note that I added emphasis for clarity in the above quotes.