The dog sounds like a great idea but I think you probably have a fox getting your birds. Coyotes travel in packs and make a lot of racket where a fox hunts alone usually. We just had to kill to foxes that were killing my free range birds. They were getting my chickens early in the morning and they were not leaving a feather or anything. It took us awhile to finally figure out what was getting my birds and I lost a good many.
Good luck with with your problem.
