I'm not sure, but I think The Teen Titans may've known.
Also, in the Pre-satellite days of the JLA, they did NOT share their identities with one another, though some members shared the info amongst themselves (Superman/Batman, Flash/Green Lantern).
Mr. Mxyzptlk did NOT know Superman had another identity as Clark Kent, as shown in the story where he changed the gender of everybody on the planet (there was a Clara Kent, but no Miss Mxyzptlk, and that's how Superman knew Mxy was behind his troubles).
aha!
Side note: I guess the Atom was the only JLA member who could travel to the cave-quarters in "pre-Satellite" days, via land-line I guess.