So, granted I am not following the OP's rules, but here is the change I would make to DSC as far as having it be as close to what we saw as possible: simply have it take place post-TUC instead of pre-TOS.
Now the sub-changes:
1. Have T'kuvma and his followers be opponents of Azetbur's government, who fear that aid from the Federation will make the Klingon Empire weak and turn them into the Federation's lackeys.
2. Have the role of Sarek replaced with Spock, and have Michael not be Spock's adoptive sister, but the adoptive sister to Spock's son _______ (you fill in the blank). Spock already had a history of being a mentor to younger people (Saavik, Valeris), so a human raised by Vulcans as another ward wouldn't be an uncommon thing. So now with Michael and _____, we have two brand-new characters with no canon backstory to have to adhere to.
3. The trip to the Mirror universe wouldn't have needed the subtext of "we have to classify it because it was new to Kirk and Spock in TOS when they found it" stuff. Its existence would already be known to any Starfleet crew, just like it was known to the DS9 crew when they crossed over.
We already know that even after TUC, there wasn't lasting peace with the Federation and the Klingons until after the Narendra III incident in 2344, so DSC could conceivably take place at any time between 2293 and that date. Most of DSC's ships and tech could fit right in to a post-TUC setting as it is, and the difference in Klingon ship design would have been less drastic. Sure, the look of the season 1 Klingons would still be problematic, but it would have been an issue no matter what time period the show was set in.
There is absolutely nothing that gets in the way of Michael being the adopted daughter of Sarek, no matter how you look at it, for example, Sybok was never mentioned until Spock was forced to justify his apparent betrayal to Kirk. That makes me suspect the people that are so insistent that there is, of having an unmentionable agenda.
And since Sybok was never mentioned or heard of again after STV lends credence to the idea that perhaps creating heretofore unknown siblings out of thin air is not the most popular thing in the world to do.
And there's no agenda here. People either hate change, or deal with it and adapt to it. Very few people ever unequivocally like it.