Indeed, the fact that the doctor is sealing a demon is a major issue. The priest will surely suspect that the doctor might be possessed by the demon. While the priest is indeed connected to angels, the power of the Death Angel is extremely dangerous after all, so it's quite natural for the doctor to suspect the priest as well.
Even if both are willing to share their secrets—which is unlikely—the priest would find it hard to believe that the doctor could constantly control that demon, and the chances of the doctor trusting a priest who holds angelic power are something Shard can guess.
Without Shard, the doctor and the priest would have to constantly search for demon or angel remains by themselves. Combined with the matter of The Chosen One, their relationship would eventually turn completely adversarial. It's not impossible for situations to arise like, "I'm helping you, Bill, put down your gun," "Priest, why don't you trust me," and so forth.
