There are enough potential paradoxes with time travel in Trek (or in general) that we don't need to invent more of them.
...I figure the odds are about the same as (in
Mirror Mirror), haiving the Mirror Universe Kirk, McCoy, Scotty, and Uhura just "happen" to be beaming up from the
exact same planet (Mirror version) at the
exact same time as our Universe's Kirk, McCoy, Scotty and Uhura. Think about it -- events in the Mirror Universe transpired in such a way that the Federation is evil, but the eight of them happen to be beaming up at that exact same moment?
If you think about the odds of any individual person EXISTING AT ALL in the Mirror Universe (let alone doing the same thing at the same place at the same time), it is quite clear that the laws of causality cannot operate the same in both universes.
Imagine if you and your spouse have a child (who has half of his DNA from each of you). Then imagine that you must go back in time, to about 10 months before your child was born, and you must re-create the child's conception. That means the child must be conceived at the same exact second, by the one sperm out of millions containing the child's same DNA, fertilizing the same egg. That would be an impossible task even if you traveled back in time with a dozen microbiologists and fertility specialists.
Then imagine if you got back in time, and found a war going on, and everyone in the universe had a different personality, and many people were being assassinated or tortured to death.
But somehow, not only would you and your spouse conceive the same child at the same moment, but 100 billion other people all over the universe would also have the same children with the same DNA over the course of 400 years on hundreds of planets, so that the same people are born in every generation, despite the fact that the history of every planet and the personality, vehicle, and clothing of every individual person in the galaxy is different. And those same children would grow up and find the same mates, and again conceive the next generation of children at the same exact moments with the same exact genetic code.
Think about it: If Kirk's parents conceived a child one minute sooner or one minute later, they could have had a daughter instead of a son. That's true of the hundreds of billions of other people in the universe, in every generation, from the 20th century to the 24th century.
Yet, in the Mirror Universe, every person is conceived by the same parents at the same moment, and grows up to meet the same partner and again have the same offspring at the same time, despite all of history being different in both universes.
We have seen that both universes have a Zefram Cochrane, and a Jonathan Archer, a James Kirk, and a Benjamin Sisko, in four different centuries. The genetic likelihood of that happening in two universes with wildly different events, personalities, and histories is approximately zero.
So there must be some force, besides causality and random chance, that somehow keeps both the Mirror Universe and the Federation Universe synchronized with one another at the genetic level. This makes the Mirror Universe unique among alternate realities in "Star Trek," since it is not just an alternate history branching off from another, but every person in each universe remains the same in every generation, at least until reproductive age.