I've found a time frame for Boureston's, but that's about all I can find. The following is from
jassdancer.blogspot.com:
Boureston's is legendary among the greatest dancers to ever step foot on a floor in Los Angeles. Between the years 1943 and 1945, the weekly dance was the hottest location for the best to have their showdowns.
It must've been mentioned in the newspapers.
I've just noticed that their very similar matchbook has the phone number Axminister 9090.