I owned a dog for 13 years and I can certainly relate to the pain you feel over his plight and his pain. However, we as humans have obligations toward the animal kingdom and a primary obligation is to alleviate pain to animals. In this case, putting down the dog is the only way to end his pain and is I believe the moral thing to do.
You are correct, we are "playing God", but that is precisely what God enjoins us to do. He wants us to exercise authority over the natural world in a benevolent, ethical fashion and He has appointed the human being as the most powerful of His creations. Your dog is very old, and I am sure was loved and cared for his entire life, he is in pain now and the most loving thing you can do for him now is to put him out of his pain.
