Heck with it. There really is no better way to fight demons than with holy water. It's a staple of the genre. Since that would make it too easy to find all demons in the Church, the AC is forced to use humans. He does, however, use an occasional demon when he needs to replace someone in a hurry, like the chief exorcist. The demon bodies will only last a few months, which is another reason they're not practical as human replacements.
Now I just need a reason why some of the clergy that the detectives hunt are short-lived demons. As currently written, Father Fischer is human and jumps to his death because he's afraid of the Antichrist. If he's going to jump, I'd rather make him a demon. Demons are obedient and expendable, so Father Armani would have no problem ordering one to jump to his death. In fact, he might have attempted to kill Campagna and De Rosa by landing on top of them, but simply missed by a meter.
The other option re Father Fischer is that he's human and found dead when the detectives get there, killed by a demon summoned by Father Armani.