I apologize... I completely missed the fact that it was a BB gun. I read that, but my brain went right to, "NJ considers BB guns firearms - so it's no different". So yes, the 4473 becomes moot - but that doesn't stop NJ from requiring it:
2C:58-3b(3):
Prior to a transaction conducted pursuant to this subsection, the retail dealer shall complete a National Instant Criminal Background Check of the person acquiring an antique cannon or a rifle or shotgun. In addition:
BB guns meet the definition of "Firearm": "It shall also include, without limitation, any firearm which is in the nature of an air gun, spring gun or pistol or other weapon of a similar nature in which the propelling force is a spring, elastic band, carbon dioxide, compressed or other gas or vapor, air or compressed air, or is ignited by compressed air, and ejecting a bullet or missile smaller than 3/8 of an inch in diameter, with sufficient force to injure a person"
and they also meet the NJ definition of "shotgun": "or any firearm designed to be fired from the shoulder which does not fire fixed ammunition". A BB gun is a firearm, as defined above, and it does not fire fixed ammunition (that is, a self-contained cartridge). I'm sure this was originally meant to cover things like Blunderbusses and the like, but it's also a catch-all for air rifles and such, too. Much like NJ doesn't have SBRs, but a 14" barreled AR is by NJ definition a "sawed-off shotgun".
So... regardless, a transfer has to occur at an FFL, receiver must have an FID, CoE must be filled out, and the FFL will just roll his eyes at the part of the law that requires a NICS check.
New Jersey... obfuscating gun laws since 1966.