I've been in every room and even unlocked the door to head towards the monks. There aren't any monsters left in any rooms, but Barud still thinks I haven't cleared them. Do I have to the caves again or am I missing a room somewhere? I've been into all of the doors and the room through the doorway at the bottom of the caves map (with the trolls). What am I missing?

There's a bug with this quest. Is your game version 2.01? If it's the older version, sometimes Barud will not acknowledge you have killed all of them.

To get around this you need to install v2.01. Else, you have to load from previous save, first go down get the key to outpost. Talk to Barud and get the quest. Kill all monsters, come back and talk to Barud again. Now he should agree with you that you killed everything in river caves.