Right, here's the elegant, less-likely-to-backfire mass-roleclaim plan.
This night, the nurse should target Demagog. Not roleclaim to demagog. Target demagog. I know he's Paranoid Mechahitler, but it's a follow-the-cop and it's worth the risk.
The nurse is confirmed town, just like the LSF, and so there's no chance of mafia being a bish.
First day, Have demagog point out the second LSF, who we then mutilate to confirm that demagog is telling the truth. We only need one, after all, and it confirms that demagog isn't Paranoid Mechahitler, which I seriously believe he is.
If demagog's target flips town, THEN we mass roleclaim. Proceed to channel all the information from investigative roles to him.
Yes, we're spending a day to make sure demagog isn't being weird, but I think the caution pays off. It's not a case of if mafia is stupid or not; I assume everyone plays optimally, and I would happily sacrifice a mafia member to avoid follow-the-cop if I was of that faction. It's a gameplay strat that kills mafia very easily and is difficult to respond to, especially given how many investigative roles we have 'ere.
Snipping off the second, unrelated section so you lot can respond to this faster.