There was an almost identical situation in my area a couple of years ago. An old man, a healthy Golden Retriever, & his wish was to kill his dog.
I'm don't remember how they got around it, but a HEAVENLY local rescue group I work with put out an emergency email to help save the dog. They had numerous responses immediately, & the dog is now living with a family (and the rescue group is VERY careful to match the pet with the perfect family).
I understand the mans wishes, but sometimes there are people who will love & take care of the pet.
I hope it works out for the dog...God Bless.