This bishop and the staff in her faction seem to be quite competent. I wonder how she ended up being appointed, and how while newly appointed she's already gathered some competent staff, and I also wonder who the other factions are who would be able to get in her way. I guess we'll see through the world building at some point.
Since the attire appears to be chosen by the person themselves, I guess that the bishop is wearing that veil by choice rather than because it's part of a bishop's attire, and I guess it's to hide that scar which we only saw in that one frame. It's a bit surprising that someone who has been disfigured in the face is in a high position for the church of that goddess, but then again, the object of their worship is described differently than how the goddess actually is, so it makes sense that a scar would not be too much of an impediment for someone who has the necessary skills to navigate the politics of the church and rise through the ranks.
It's a bit weird that the priest introduced himself to Kuzunoha with the request of the "old geezers" but he's apparently not a part of their faction and is instead in the bishop's faction. I guess it makes sense that Kuzunoha was directed to the bishop rather than the "old geezers", whoever they are, since they request was flatly rejected. Still, they can't be too happy about it though so I'd expect some movement from their side, though it seems they've done nothing for now.