You wouldn't use a Form 4 to register a firearm in the NFRTR as a Form 4 is only for a "transfer" of an existing firearm. Form 1, 2s, and 10s can be used to make and register a Firearm.
If this theoretical Pistol brace amnesty works like the previous Streetsweeper, Striker, and USAS Amnesty you would not be required to engrave the firearm as you did not "make" it the OEM manufacturer (SWD, Sentinel, or Gilbert) manufactured the firearm and you are just "registering" it via a non-tax paid Form 1 due to a reclassification. If you are an FFL/SOT 07/02 you may use a Form 2 vs. a Form 1 as well.
I can say that my individual amnesty registered Striker-12 doesn't have any additional engraving and the Form 1 that was used to register it lists Sentinel Arms as the original manufacturer.
The ATF did track down as many sales records for these shotguns as possible and attempt to proactively notify their owners of the reclassification. Per the ATF updates when they closed the amnesty in 2001, they reported to accept ~8200 registrations out of ~17,000 guns between the three types that got reclassified. I never heard of the BATFE going to the doors of people who they got a record of ownership via the 4473 but that person never registered the firearm. Technically you didn't have to register, you could just remove and permanently separate the barrel from the receiver, destroy the firearm, or surrender it so just because you didn't register didn't mean you were in possession of an illegal DD after the amnesty closed.
Whether that is how a proposed Amnesty Pistol Brace program would work and would they require engraving on a factory braced pistol (or any braced pistol even if assembled by an individual from parts), will you use a standard Form 1 or will they come up some some new amnesty Form 1 doc, etc. is all a complete guess at this point.
For braced pistols, they may very well ask for assistance from OEM manufacturers who sold factory braced pistols, but given the ATFs propensity for mass copying 4473 they probably know the vast majority of folks who bought factory braced pistols. However my guess is that less than half will get registered and the majority will just remove the brace or update the brace config to one that doesn't meet the size/weight/points requirement to be an SBR. So going and kicking in half a million owners doors who didn't register is probably just not practical or legal when there are so many other options to comply other than registration.