No, animals don't FEAR the pain of dying (and don't fear dying, for that matter), because they live in the moment, so they are only fearful when actually in the process of being scared (and they don't have a belief system that registers the end of life or moving to another cycle like we do, so they don't even conceptualize death like we do). They don't anticipate things, they don't think ahead like we do. But, they do still feel pain, and would rather NOT have to feel it. Their pain responses are just like ours, if you hurt a cat, it will pull away or vocalize. They will run to avoid pain. (ever see one move it's tail just as you were about to step on it?)
I think the AC's you've spoken to are wrong. Phaewryn I used to think this too but since I've heard from many AC's that animals don't fear the pain sometimes associated with dying in some instances and many actually prefer to go through the whole cycle I am really more of the thinking of letting them complete the cycle without my intervention. Now if I clearly get the idea that an animal would rather not go naturally I will help them and having done it both ways there to me is no harder thing than helping an animal go, although the other is pretty tough also.