You could easily pull something like
Scion of an Vanquished Religion: Only affected by a dedicated weapon of a major existing religion i.e. Sword of the Cross, Researchable
That's a +1 Catch that will never be satisfied as long as you remain one of the good guys.
Hello, free point of refresh.
It would be like a player having the aspect "Likes to Sleep" and self-compelling for a fate point every time they wake up.
As a GM I invoke one of the core tenets of Hero System in any game I run: "A disadvantage that isn't actually a disadvantage isn't worth any points."