If you could ask the chicken, what do you think it would say? It is very hard with our pets as they become part of our family, and we put our human emotions and ideas upon them. But the reality is, I bet that chicken wouldn't want to carry on the way it is. Nature would usually sort this quandry out very quickly, and the chicken would perish, and animals accept this 'order' of things without thought. I feel if we keep animals, we have to be prepared to do what is right for the animal, putting aside our own emotions and needs.
As for your wife, I agree with the other members, that communication is the key. Hopefully she will also come around to seeing what is best for the chicken, and that she will feel reassured that at least the bird will be put down quickly and humanely.
best luck!