I once tried to figure out what TOS episodes could be retold as Mirror Universe stories, and the answer was, not many. Generally, they seemed to break down into three categories: stories that wouldn't happen at all because they were about humanitarian missions or freeing oppressed societies; stories that would be wrapped up very quickly by just blowing up the problem; or stories that would unfold pretty much the same because they were about fighting off attacking monsters or doomsday weapons or what-have-you.
This reminds me of one of my favorite Trek board games from the 80s: Star Trek: The Adventure Game by West End.
A federation player and a klingon player fly to different planets and experience short choose-your-own-adventure style "missions" exploring each planet.
Most of the missions were loosely based on various episodes of TOS. (And other things - one was based on Forbidden Planet, which was extraordinarily cool.)
Anyhow, the Klingon missions often played like "What would the Klingons have done if they had experienced this TOS episode instead of the Enterprise?" Most of them were like what Christopher described - they would just blow up the problem, or intimidate people.
Its not the mirror universe Federation, but its pretty much the same idea.
Man, I need to dig that game out of my closet again, I bet my kids would love it.