Very interesting, though I think setup 2 may be a bit more town-favored than setup 1. I wouldn't even bother asking for a claim on a lynch, a cop that's outed day 1 is useless and it's not like they can be counter claimed.

They are not useless. Because if they have a partner, the partner knows he's sane cop. Also, mafia needs to kill him. But if you lynch the cop claim, means mafia can kill one person hook the another one with higher chance of hitting/hooking second cop.