I don't see where in the US Code the ATF could require serializing and registering the
"pistol brace", rather it'd be the receiver.
The original rulemaking request was titled
“a proposed rule to make clear when a device marketed as a stabilizing brace effectively turns a pistol into a short-barreled rifle subject to the requirements of the National Firearms Act.”, which appears to acknowledge that devices to turn pistols into short-barreled rifles are not regulated items in 18 U.S. Code § 921 -- for example,
While the law around AOW, machine guns and destructive devices (
26 U.S. Code § 5845) and also silencers (18 U.S. Code § 921) have language such as "
including any combination of parts, designed or redesigned, and intended for use in assembling or fabricating", no such phrasing exists in the SBR/SBS/AOW law, thus the ATF has no authority to do rulemaking requiring registration or marking of SBR/SBS/AOW "parts"
They can still
claim that possessing a "brace" at the same time as an AR pistol is equivalent to possessing a SBR, but the brace itself isn't a controlled item subject to registration.