Keith,
The answer is no. Your FFL dealer does not have to enter the transfer in his book. The book is required only for firearms and this shotgun, in the eyes of the BATFE, is not a firearm. Since this isn't a firearm, no check is required either. I know this because I just picked up my FFL, after months of hassle with the town board, so it's still fresh in my memory. To back this up, I've got a friend who's been selling guns for years, and is also a lawyer, who has instructed me along these lines as well.
I would ask my FFL to simply send them a copy. Don't bother entering it at all, or enter it with a note stating it's an antique. At that point he has no further requirements according to the law. I doubt you'll get the seller to back down, just from past experience.
Luck,
Jim