Nice try, and full marks for creative thinking. However, there are a few problems that you need to be aware of with your line of reasoning. The first is the way in which that the only person who can hear this strange noise at the end of this grim and gruesome story is the narrator himself. The men who are with him and investigating the scream that was heard continue to "chat pleasantly" and smile in spite of the deafening sound that drives the narrator to such paroxysms of rage and frustration. If it was a watch that he had left in the room, these men would have heard it too.
Secondly, let us remember that Poe gives us every reason in this narrative to doubt the sanity of the narrator. He is presented as being profoundly unreliable, from the grandiose claims that open the story as he claims to hear "all things in heaven and on earth" and "many things in hell." His obsession with the old man's eye and the subsequent way in which only he is shown to be able to hear the sound indicates that this is yet another symptom of the narrator's madness, as he concludes that the sound he hears is "the beating of his hideous heart." However we interpret this sound, whether it is some remnant of the narrator's conscience that causes him to confess, we can definitely assume that this sound is imagined in the narrator's head.