To obtain this reward after fighting the mercenaries at the waterfall, first pick up all the dropped items you want to keep for the upcoming boss fight. Then go through the doorway where the white-shirted guy was standing. Cross CHECKPOINT 20. Shoot the leopard in the next room if you want to or wait and deal with it later. The important thing is to use the checkpoint. Then if Lara dies while going after the reward, you won't have to fight the soldiers again.
At this point, I recommend swinging across the lower set of horizontal poles to re-use CHECKPOINT 19 at the entrance to this area. Then swing back to the corner column and climb back up as you did before. Return across the wooden bridge (or ropes) to the exit and go on. Pick up the walkthrough at CHECKPOINT 20, which will register again if you've used CHECKPOINT 19.