The fact that Lucifer, even without any pretense looks to be the same as any other person in his "flock" really brings to mind something that is pretty big in Faith (banger game btw): The Devil himself would not come as a monster, unlike every other demon. Much like Imuri, he'd show up as what little Priest wants the most. And immediately he gets to show his power of resurrection. Considering he seems to have omniscience since he was closest to the Big Man Himself, he might have timed the Dog's lifespan to the second, just so he could show off to Priest.