The World is Weak
Tick off Jacob enough, and he'll send his hunters for you.
As with the other boss-story missions, your capture here is unavoidable. Jacob's archers will shoot you with poison arrows, and you'll wake up in detention.
After the show & tell portion is over, you'll be thrust into a kill-or-be-killed situation with the other captives.
The Deputy doesn't know any non-lethal takedowns, so you're pretty much going to be killing these people. If it helps, this is at least partly a hallucination. It's also a great place to score some sidearm takedowns (though any perk points you earn this way won't be awarded until after the mission completes).
If you die, you restart nigh-immediately, Miami Hotline style, so feel free to cut loose. You'll start in the filmstrip room, with a pistol on the table. Take out those two guys (though you can run past them), and run out to score an pistol SMG (though you don't have to take it).
The next section has four guys in a stage+stairwell area. You can vault the stage to take them out head-on, or go all the way left (past the tv) to flank them through a doorway.
You can use the same doorway to dash/sneak past these guys if you'd rather. Past their section is a shotgun.
The next section is hotel-like corridors. Take the left branch. Baddies will be hidden in little dead ends, but push, punch, or pew-pew past them. A giant bloodstain will indicate the exit to your left, along with an assault rifle.
This next and last section is the hardest one to clear pacifistically- you'll find yourself in the cross fire of multiple enemies in multiple positions, around a fountain.
Run and gun through here for another corridor with another weapon pickup.
The next & last stage is more stairs, but these ones are broken, and mildly on fire. Take the stairs up, jumping gaps where required.
At the stairtop, look for the abyssal doorway.
Go down this passage for a long cutscene, which eventually ends with you recuperating in the local militia base. This completes the mission.
There's an arcade machine here, as well as 40 silver bars by the boombox.
Follow your marker and talk to the not-entirely-pleasant locals to get the Missing in Action, Gearing Up, and Radio Silence missions.
You'll also trigger the Mountain Armed Convoy side mission.