I'd think base board locations only would be the way to go. An associated herald that forces you to close an expansion neighborhood would be interesting, though it would obviously need to do something else so that it's actually harder. I do kind of like the idea of being forced to choose between a world-destroying rift and the hospital/asylum, though.
I could always have it add a doom token (or maybe close another neighborhood instead) when one of them kills an investigator if you're worried about it being used for that tactic.
Or perhaps draw a random monster trophy and put it on The Blessed One.