An early story outline for "The Alternative Factor" sent Kirk to the anti-matter Enterprise, where he eventually meets the "good" Lazarus, after having a few weird interactions with the alternate Enterprise crew. I doubt Bixby was aware of this, however.
And Roddenberry's original 1964 series prospectus included among its "story seed" premises "The Mirror," in which the starship Yorktown came upon its exact duplicate, but only one of them could survive. Although that's more like Voyager's "Deadlock" than "Mirror, Mirror," I guess.
I'm sure there were a lot of SF stories about alternate worlds that were a "mirror" of our own -- in some ways, it goes back to Lewis Carroll's Through the Looking Glass and the like. DC Comics introduced their "evil twin" universe Earth-Three in 1964.