I had never seen anything like this before. I asked Larry Fuhrmann about it, figuring if anyone would know if BBM had ever done some sort of promotional item for The National, it would be him. But Larry's never seen these before either.
I have to say that it doesn't look like the kind of promotional item BBM does in Japan. Their items usually are new cards with the same picture but different colored graphics - like these Lions promos from 2013. Or the promos in SCM or Weekly Baseball. David said there's nothing stamped on the back of the card.
I did a little looking around on Ebay and I found a bunch of 1991 cards with the same stamp being sold by Burbank Sportscards. David mentioned that he'd seen some being sold by other sellers but I haven't been able to find any. Burbank Sportscards is currently listing 353 cards and there was only 399 in the original set so assuming there really are only 1 of each of these (and taking into account the Hasegawa card that David bought) that's only leaving 45 cards unaccounted for. Obviously it doesn't look like these actually got distributed at the National if one seller has possession of 88% of the cards.
My gut feeling is that someone other than BBM picked up a 1991 factory set and thought it would be a good idea to stamp each of them as a promotion for The National. Since there's no sponsor indicated, it may have been the promoters of the show itself who did it. I've sent a message to Burbank Sportscards asking for details and I'll update this post with their response if I get one.
Any information anyone has on this would be appreciated.
UPDATE: I got the following response from Burbank Sportscards about the cards:
Thank you for your inquiry. Unfortunately, I don't know when exactly they were given out, but I do know it was sometime in the mid 2000's. As for the company, I believe it was a company that sponsored the shows.