The breathing animation works, but the rifle will do its transformation animation at the beginning (ever since linking it) and there is no indication where to override this since the bones don't have any keyframes. Did I miss something somewhere?
Edit: Ah the next part explains the NLA editor. There I can see the rifle_transform action. Should I delete this from here?
You're right that the secret is the NLA editor! If you delete the action it'll delete all your hard work (unless you clicked the shield icon in the action editor), so I wouldn't do that, but you can move it around so that it comes after the breathing animation or whenever you want it to play.