First of all, I never believe in the concept of kiling a living soul to end its suffering. I believe every type of life is precious, human life, animal life or even trees for that matter.
To end a life from suffering is like saying if you suffer from terminal cancer, I'll run you over w/ my car. I think that decision can only be made from the person who's suffering.
Since a rabbit can't talk, there is no way of knowing that rabbit wants to end her own life.
And I found it absurd that a diagnosis can be done over the phone. This vet should be fired as he has no caring about animal life.
I always thought if there is an animal that's suffering and w/o knowing it is in fact dying, the least we have to do is to give that animal the emotional care that we can provide.
I'll bring it home w/ a heating pad to warm up the rabbit, w/ syringe to feed some water. That way, even if the rabbit's sickness is terminal and there is no cure, I know and the rabbit knows she died peacefully w/ someone that show love and care, and take cares of her final moment. Psychologically speaking to the rabbit, I found that to be very important.
I remember 10+ yr. ago, one of my bird was sick (and turned out she passed away shortly after) and lying at the bottom of the cage, the other bird fly down to show his love to the one that is lying at the bottom of the cage. So I learned back then, even among animals they show compassion towards one another.
As to the case in hand, why not let the rabbit sit around in the house, nice and warm and bring it to a vet the next morning? Knowing the vet is clearly an uncaring person, I wouldn't be so sure that his PHONE diagnosis is right to begin with. What if it's not myxi, what if it turns out to be something else entirely? And even if it is, say if the outcome of letting a rabbit stay overnight allows some vet treatment the next morning and prolong the life for even a few days, I'll take it.
Sometimes, little animals in the wild has certain chemical in their system to release itself to end its own life if they believe it is necessary.
I once took care of a sick sparrow in my backyard and she was being killed by a magpie because she was sick to begin with. After I intercept it, and let the vet took care of the sparrow, the bird has some sort of self release chemical inside her body, and the bird must be ignited that chemical and terminate her own life peacefully.
Have you ever heard of any doctor making a diagnosis over the phone and make a judgement saying, don't bother come down, just end the patient's life? But having said that, it took a lot of courage to write the story like that, and for what it is worth, this story should give us the experience to learn what the right thing to do would be if it happens in our life in the future.