FWIW....

The August 1956 phone book lists

Eddie's Bar-B-Q at 86 S. Maine
no listing for Eddie's Club

it lists these clubs on S. Maine:
Bank Club 158 S. Maine
Clover Club 46 S. Maine
Club Horse Shoe 138 S. Maine
Esquire Club 20 S. Maine
Keystone Club 168 S. Maine
Owl Club 50 S. Maine
Palace Club 60 S. Maine
Sagebrush 98 S. Maine (o.k. it doesn't say club, but it's in the area!)
Star Club 70 S. Maine

Not likely that any of these share the Eddie's Bar-b-q building...