I still don't see the point in insisting that the facility be a real and preexisting one - after all, the surface construct used to describe it is a set rather than a real location. At some point, the mimicry is going to break down. Why not forget all about it from the get-go?
We aren't really trying to claim that the McKinley rocket base of TOS fame would be Kennedy Space Center by a different name, even if in the real world Kennedy was the only place in the universe capable of launching a Saturn 5. It's not the same universe. The Trek universe Earth in the 1960s simply had more and better spaceflight facilities; it probably had more and better nuclear missile launch facilities in the 2060s, too.