I wonder how much business this one brought in. Really: if you didn't one to go to one club, chances are you wouldn't make it to the other for closing time.
But I'll bet TWA was the co-sponsor.
The Stork Club was a famous swank drinkery for the smart and connected, bur not everyone; Bogart was banned, as well as Gleason. (Alcohol, as the police say, may have been a factor.) It's a fascinating story of a time and place long lost, and the wikipedia entry is a good place to start.
The Troc was the place for Hollywood types in the 30s. Bogart could probably get in there, no matter how many loud fights he had with his wife.