I've heard conflicting things about this... that the lower has to be declared a pistol or rifle either from the manufacturer or on the 4473 itself during the transaction.
Since an AR lower can be built into a pistol, I guess it makes sense. I don't agree with it, but it makes sense. The whole SBR and pistol nonsense is a legacy holdover from the 1934 NFA garbage and really needs some re-examining.
Especially in light of Miller, combined with Heller, and the fact that a huge percentage of our Armed Forces uses a sub-16" carbine.