Fair questions. A priest is no different than any other witness, however. It should be up to the jury to decide if they believe him. Proving crazy to a jury would seem to be no more doable than possesion, doctors give opinions based on their examination, experience and training, same as would a priest. All the defense has to do, at least in theory, is present reasonable doubt to a jury, not prove possession.
My guess is most lawyers would advise their clients to take the mental illness diagnosis to save their lives, would probably take a devout attorney to press forward with the possession defense at that point.