I also note there doesn't appear to be any distinguishing factor that allows you to win better than chance odds, but I haven't actually played a lot of Mafia before, so I may just be unfamiliar with the strategies involved.

Well, it's usually played in person, and humans (usually) aren't perfect liars.

Your proposed game has one flaw - there is an FAI and they want to help you win. It might be closer to have only two players, and the AI flips a coin to decide if it's friendly - but then they would win if they let it out, with 50/50 odds, which seems unrealistic.

Perhaps the AI decides, in character, after being released, whether to be Friendly towards the human? Then the Gatekeeper could try to persuade the AI that Friendliness is optimal for their goals. The temptation might help as well, of course.