SilverWook said:

The TOS footage could even be windowboxed, (ala the tv scenes in Galaxy Quest when it was shown in theaters) then you open up to the full theatrical aspect ratio.

I think windowboxing would make it even more obvious that we were reusing unrelated scenes to tell a story. Cropping it to 2.35:1 might make you wonder where the footage was found a bit longer.

Another scene from "This Side of Paradise", where a smiling Uhura informs Kirk she's disabled communications, and then leaves the bridge might also work in that montage. :)

You wouldn't wonder though. You're really good at this. I'll add it to the mockup.