If we are willing to set aside the real world elements of a murder investigation, there are a couple settings that might work.
Modern Industry
Moderator is the CEO of a company and he knows there are spies among their employees because there are verifiable data leaks within the systems, but he could not track where it comes from. The company dabbles with barely legal activities regarding finance/hygiene so they don't want the police involved, but they still want to remove the spies. The CEO does not have time to confront each employee directly, so he just asks that they sort it out by themselves and if the things works out good the loyal employess will be rewarded - those who are erroneously fired in the process will be hired again.
So, each day the employees gather in the cafeteria and try to identify who's a spy, producing a file for the CEO who will immediately fire the accused person. The company will notice if the data leak sources have been reduced or not, which is proof of the fired guy's alignment. During offtime, the spies gather and attempt to delete the hiring contract of one of the loyal employees to sabotage the system. They cannot remove too many otherwise the company will collapse before they could steal all the relevant data, but their first and foremost priority is not getting caught so they will have to react accordingly to the ongoing crisis. The spies will complete their mission once they have gathered enough data to send to their contractors and have enough presence in the company to take it over, which occurs when the spy population reaches 50% or more of the company workforce.
All of the involved people are human and civilized, none of them want to involve the police, so they wouldn't resort to more efferate crimes to achieve their objectives.
New Religion
A group of people is trying to found a new church, in direct opposition to the already existing religions. People in the village do not accept that, so they run to the moderator (the pope or something) and ask him to solve their problem. Pope could not give a damn about it, but he has a duty to listen to his subjects otherwise they would revolt. He asks the villagers to find who is a member of the new religion so that he can exile him at the end of the day. The pope will run an investigation of the exiled person activities to determine his alignment, but other villagers do not trust the guy anymore so he will not be called back until the problem is solved. During the night, the insurgent fanatics kidnap a villager and try to brainwash him privately, but they still need him alive or he won't be a worshipper of the new true god.
If the new religion becomes predominant, a new church is publicly founded and the village is officially converted. Exiled worshippers are called back and have their share of the victory. If the new religion is eradicated, exiled loyal followers of the pope are called back instead.
This is the best i could come up with so far, it's hard to create a consistent system where common logic can be safely ignored - such as why the good guys cannot investigate first and kill later, or why the bad guys cannot kill more than once per night - as well as letting players enjoy their victory, instead of having to die to see their wincon fulfilled and therefore leading to the paradox of "winning posthumously", i mean, technically if you died you can no longer win anything, even if your faction achieves your wincon later.