Beyond the recent info about the promo, there are a few other things about the cup that mark it as modern.
The font for "Briarwood" is a modern one that wouldn't have been used in 1923. Also, BBC Co. would not have referred to itself as simply "Brunswick". The very plain design of the Briarwood side of the cup is not one that would have been used in 1923.
In 1973 Brunswick printed a reproduction of the 1923-1924 catalog. The image on the front of the catalog is the same as the image on the fancy side of the cup. Here is the image that was on the back of the catalog reproduction:
I vaguely recall "Briarwood" as a Brunswick brand from years ago. It is a modern line of Brunswick tables that seems to have been discontinued. You can find it through google but it is not directly reachable on the Brunswick site. Here is a google-found image:
OoooOOOOoooo its all the same cup....now I see.
At first I thought we were looking at two different cups.
Yeah the Briarwood logo was definitely not from 1923...my bad...