I can't recall a singe instance but I am looking to disprove my hypothesis that since truly adopting the 12th AoF at the beginning of the 20th century, the church has never spoken out against government-sanctioned injustice.

Prove me wrong!

(Maybe not the best forum to pose this question...)

...they don't want to risk having missionaries kicked out, so rather than standing for something, they turn a blind eye.

Besides, the church is so self-obsessed it's probably oblivious of injustices. And in the odd instances they become aware, they probably think the solution is to convert more people.

Not that I recall. It mostly just decries injustices against itself.
As for the problems of the world, the basic stand has always been that if they had the LDS gospel, those problems would take care of themselves.

And remember when Clinton was having his troubles, and Hinckley was on tv? He simply refused to say anything bad or condemnatory, but that the laws would take care of it. At the time, we were told to think that it was wonderfully generous of Hinckley, but I suspect that he was really just trying to focus of keeping people from getting mad at us. And that has always been priority #1- to be as popular as possible in the eyes of the world.

Let all hell boil over and let calamities arise as long as they do not negatively affect the church, but once they do, we'll cry persecution as loud as we can.

Nazi Germany, for instance: The church likes to spin stories about certain German members who stood resistant to the Nazi regime. Yet during that time, the church sucked up to them as much as any of the other churches in order to preserve their status.