This happened to me this year with a deer, and I tagged it. The WCO's have the option of issuing you a new tag when this occurs in the state I was hunting so I figured why not tag it. Plus I did kill the animal as your wife killed that turkey. You wrote
My wife cried over the whole mess.
I'm guessing it wasn't the killing of the animal she was upset over, or she wouldn't have launched the arrow ... what I am guessing bothered her was the leaving an animal she killed in the woods to become worm food, which you could have made better by tagging it and taking it to a WCO and telling your story and hopefully getting a new tag. The fact your wife cried over this meant her conscience was hurting .. so why not tag it, if for no other reason then to make her feel better about the experience.
Not saying what you did was wrong, that's just my opinion.