When I was about 6 years old, my cat Grettle got run over by a car. Her face was busted, she could not walk well, and was in a fair bit of pain to say the least.
We took her to the vets, they said it would be best to put her to sleep. I said no, no way. Instead they treated her, gave her pain killers, and fixed her up. After about a month or two she recovered, she went on to live almost another 20 years, died in her sleep at the grand total age of 24. (give or take a month or two)
I say go with option 1 and remove the leg.
Before anyone goes crazy at me, allow me to defend myself on this one. I think it is the best option, as it will combine the minimum “force” on the dog, and most ethical choice. You are not imposing so much on it by taking the leg.
With option 1, you impose a missing leg, and the chance of a death that would follow soon. Obviously this is not good for the dog, but there is a chance that it could recover, and it could indeed have another 2 or 3 years in it. Honestly, probably not, but it could. In total, with option 1 you impose pain, a chance of death, and a chance of life.
With option 2, you impose pain, and death. This option has no up side, and has more negatives than option 1. In my opinion, this means it does not beat option 1.
With option 3, you impose death. You could try to claim that you prevent pain, but you don’t. The dog will not enjoy being killed, however humane it may be. There are many regulations about the execution of prisoners, yet you still hear horror stories. You can say option 3 does minimize pain, but it has no up side, and it imposes the harshest option, death.
Yes, the poor guy does not have long left anyway, yes I know he will suffer a bit with surgery, but it is the more ethical option to me.
Dogs have no concept of death, so it wont feel the terror a condemned man feels on the long walk, but dogs also don’t have a complex understanding of suffering like humans do.
If we assume your dog is a human, who has been turned in to a dog with a magic spell, the best thing to do in my opinion, if he has lost the capability to communicate with humans, is to assume that he wants to live.
If he could talk, what would he say? would he go all spiritual on you and tell you everyone has to leave some day? or would he tell you to crank up the pain killers and bring him a bitch a bone and a good head scratch?
If you think he is ready, and he has had his time, and that he wants to go, then let him go. If he looks like he still has some fight in him, go with the leg.