I watched DS9 fairly regularly, but I don't recall a "Mirror Universe."
Really?
In TOS (The Orginal Series, from the 1960's) there was an episode that was titled "Mirror, Mirror". There's a transporter malfunction (naturally) and Kirk, McCoy, Scottie, and Uhuru are swapped with their doubles from a parallel universe. In that universe the humans ("terrans") have built huge, evil, despotic "Empire". If you've seen Spock with a goatee and a bunch of the crew wearing gold lamee sashes, then you have seen this episode.
DS9 did a total of 5 episodes where the DS9 crew (or various subsets of them) end up in that same parallel universe. If you have any memory of Kira looking like she had raided Catwoman's wardrobe (skin tight, black jump suits), then you are remembering something from the "Mirror universe".
I understand that
Enterprise did a couple of "Mirror, Mirror" prequel episodes, but I never saw them (having given up on Enterprise before they got to those eps).
I agree that the Mirror Universe is not a likely place to continue the Trek *movie* franchise.
If you do it with new characters and from the POV of the "good guys" then it would turn out feeling a lot more like a new version of the original trilogy of Star Wars movies that happened to use the Star Trek alien race names.
If you do it from the POV of characters that we know then you have two choices. The first is to spend enough time establishing the "regular" universe, the characters, and then the parallel universe crossover so that even the complete non-fans (of all things SF) who just go to "event" movies will clearly understand what all is going on. By the time you do that, you wouldn't have enough time left to really do all that much of story in the Mirror Universe once you get there.
The other option is to dive right into the mirror universe with little or no introduction (like series episodes did). To non-SF-fans this will feel very odd. Everything about the trek universe will feel turned on its head (because it it; that's the whole point). They won't get it (by and large).
I really think that Mirror Universe types of ideas work much better in the context of a TV series (and particularly a long-ish running series) than in the context of a movie sequel/series. A good writer might (maybe) be able to make something interesting out of a Trek TV series that used the Mirror universe more extensively. Maybe a premise a little like Voyage only going to the Mirror Universe instead of another quadrant of our galaxy? However, it would be tough to do in a movie and maintain the revenue expectations for Trek franchise movies.