Let's assume that the Halkans are a race of universe-manipulators on the level of the Q or Metrons or whatever. They created the Mirror universe to test our crew of valiant heroes.
To populate this universe (with some sort of artificial intelligence doing all the heavy lifting), they took the genetic template of our main cast and those they worked with, and placed them in the world in an organic way that fulfilled the necessary objectives. Mainly, A) these dopplegangers exist and B) they exist in a world where the Federation (or "Terran Empire") is an aggressor state.
Now, we know from Mirror, Mirror that a Christopher Pike existed. If Kirk had looked him up, he would probably see Jeffrey Hunter's face. Doesn't that mean Pike was also genetically templated over to this universe? Maybe, or maybe he only exists as "backstory" available for Kirk and crew to read. Pike, anyway, had close affiliations to Spock, but was also known to Kirk from his transfer and the Talosian incident the previous season. It seems likely that such close connections were templated over, even if they were retroactively killed.
Now, let's look at IAMD. In the Kelvinverse, Archer is known to Montgomery Scott. This could be true in the prime timeline as well. Since Scotty is one of those beamed over by the Halkans, they may have taken his memory of Archer and constructed his crew as backstory for the Terran Empire. Now, obviously, the "backstory" exists as more than just words on a computer screen, because the Prime Tholians were experimenting with the 2155 MU. Notably, this was the 2268 Tholians *after* the Halkans created the MU. They used a future event to help establish the existence of Constitutions in the MU. Do the Halkans exist out of time? Probably.
Another person that Kirk met, admittedly after the Halkan incident (although there are no stardates in Mirror, Mirror, so the production order could be wrong), was Benjamin Sisko. Sisko and his associates are the only ones outside of Kirk and perhaps the Tholians (and that was another of Kirk's adventures) to have traversed the Halkan barrier. The fact that Sisko and Dax and O'Brien and Bashir are all waltzing around the Enterprise within weeks of the Halkan incident, and also have dopplegangers in the Mirror Universe (in their proper times), doesn't seem like a coincidence. We have precedence for this in Marlena Moreau, another crewmember that Kirk did not know but was templated over.
If Discovery were to ever have an MU episode, it could be tied in simply by Michael Burnham's close connection to Spock, and other potential connections that may exist between the two crews. It's possible that some of the crew of the Discovery may even be on the Enterprise in 2267 during Mirror, Mirror (probably minor crew in the lower decks, but it's only ten years later). With the large amount of TOS references in Discovery, I think it's inevitable that some of the crew will one day transfer over to the Enterprise.
So, the MU can still exist as its own seperate universe and still be a creation of the Halkans, with the three main eras (Archer, Kirk, Sisko) all having personal connections to 2260s Kirk and Scott.
The only issue would be a future Discovery episode (or maybe some future series or movie) coming up with an incompatible explanation, but c'est la vie.