Jump on the Shrine to Clavicus Vile. Make sure you arent slidding off. Make a save here. Tell him you dont want to kill the dog, then exit the diolauge, jump off the Shrine and over the chest, as soon as you can; cast the slow time shout and sprint to the exit of the cave. Spam the map button as soon as you exit the cave and travel to Whiterun (Although it may be slightly easier if you have Lakeview Manor.) ( This is because during this entire process you will have a fixed camera angle). Once you spawn in whiterun spam your Slow Time shout again and sprint towards your Breezhome. Once inside simply place the Rueful Axe on the weapon rack to your right. You wont be able to turn and place the weapon at first (If you are shouting and menuing at the optimimal times) dont worry, you still have just enough time to turn and place the Rueful Axe before its taken away. Now I'm not sure if theres a time limit for this next part but I'de just reccomend you do it my way for mine has been verrified. Fast travel back to the cave, but go through the exit of the cave. From where you left, not the beginning of the cave. The exit of the cave is just up the road to your right once you spawn at the cave. You're going to have to spam the jump button and maybe strafe a little bit to get to the exit of the cave. Once inside Clavicus should be speaking to you and will soon give you his Helm.