A really important principle about cops which seems to be overlooked here is that the thing which makes them worse is the possibility that they could be insane. Whether they're actually insane doesn't matter.
Say you're the cop and you know there's a 50% chance that you're insane. Now you cop someone and get, say, an innocent result. That actually tells you precisely zero about the person's alignment, whether you're actually sane or not. Because you don't know whether you're sane, so you can't trust the result. It's only once you get more than one result that they become useful: if two people both come back as innocent, you know they have the same alignment. Or if they come back with different results, you know their alignments are different. Once some player you have investigated flips, you know whether you're sane or not. At which you're effectively a cop, sane or insane again doesn't matter.
(You can also show this mathematically. For example, you can show that P(innocent result | person is town) = P(innocent result) which implies that both events are independent.)
So you definitely don't always want to make the cop insane as scum. Actually, if the cop knows you picked it, then it becomes useless. If they expect a 50% chance that you picked it, you get the worst possible situation for the cop; namely that their first result is useless. That would be the ideal case. If the game is repeated, that means you should choose the cop thing with some probability between 0% and 50%. If it's just one game, it becomes a pretty interesting WIFOM, where your optimal choice depends on what you think the cop thinks you do. If you think the cop thinks that you choose insane with 50%, you should not choose it, because it doesn't make any difference. If you think the cop thinks you definitely don't choose it, then you should choose it, because it'll make the cop falsely rely on their faulty results.