During the task to destroy cutouts, go to the boarded-up door, and break the boards. In the door, a searcher should spawn. Kill it. Grant's audio log is also located in there. Another axe is also in there, but isn't usable. During the last task, before going into the elevator, grab the axe. It is now clickable. I normally die twice trying to kill him with the axe, but it takes about a few hits to kill him.
Choose demon path. This will make Bendy appear more, though, so be careful. Next to Shawn Flynn (Jacksepticeye)'s audio log, change the ink thing to the ink machine. Do every task without dying. Then, you should be able to receive the gun. If you are playing on PC, you also get an achievement.