Well, now that the archbishop turned out to have caused most of the Stray-related problems in the world, that makes it simple.
The best solution then is to fix the root cause of the problem. Without her or her twisted beliefs and forbidden knowledge, otherworlders aren't gonna continue to be summoned, be tampered with and cause further disasters.
Existing Strays that have gone bad still need to be dealt with though. If possible, find a way to permanently disable their cheat skills, as an alternative to killing them.