Father Brown deliberately misuses the Last Rites on victims.
He knows full well it isn't done after someone's death, but he counts on others not knowing that. He thus gets to use it an excuse to be able to examine the victims and the crime scene. "I must be allowed to give the Last Rites, Inspector." He doesn't ever do it when there's other Catholic priests around, either.
After all, he routinely uses "providing spiritual counsel" as an excuse to visit suspects in the station's cells and ask questions about the crime.... even suspects that aren't Catholic! And never mind his use of the confessional booth to converse with suspects and/or others like Flambeau. So this would just be yet another convenient Catholic ruse for him.
