I'm looking to do the Military Police route, but I think I fucked and missed it. At what point do you infiltrate the ministry and fight Asmodai?
Alright, so I figured out how to infiltrate the Military Police building.
After escaping the cathedral, head to Resistance hideout in the special political and economic zone and talk to Puma. The hideout can be found above a smaller building with a red post beside it. Go north past the red post box and you'll reach the hideout. Once inside, go up to the second floor, enter the room, go up the ladder, and you'll find Puma and Akira.
In order to infiltrate the ministry building, you have to find someone who can tell you about the security there. Be sure to talk to Akira before you leave the room.
Exit the hideout and go left to head back to other part of the SPE zone. From there, go south and you'll find an NPC sitting on a bench next to a white vending machine. Talk to them and they will give the information you need. Once you're ready, talk to Puma to begin the mission.