I agree completely, IF you were to put him down, it should be because you feel you have done everything in your power to put a stop to his current behavioral issues. Definitely not because he could potentially kill livestock. Any dog has the ability to kill livestock to some degree, depending on...
If you feel you have done all you can, then I agree that humane euthanasia is the kindest thing for him. No kill shelters could mean years of being locked up and feeling abandoned, and taking him to a kill shelter would be extremely stressful. Although, if you really did want to try an animal...
Maybe he needs an outlet? Runs, walks, playing frisbee, one on one training, agility... did I mention obedience? Without having time to read all the posts, I would think that your dog really needs an outlet. Locking him up or dumping him at the pound is not the answer. If you cannot be with him...