If the player is up for it, I’d lead them on all sorts of red herring chases and false leads. Eye witnesses to crimes would turn out to be liars; mice claiming bankruptcy would be wealthy; all sorts of con-mice would try to convince the patrol mouse to follow them on any goose chase.
In fact, that could be the life of the campaign–following untruths or half-truths to discover what about a story has a shred of truth to it and where to look for that truth.
But I’d do it with a fair degree of transparency to ensure the player is making a decision to follow thru on the PC Belief or playing against the Belief according to choice, rather than chance. What I mean is, if you use the red herring or goose chase too often, the player might begin to think it is always the case and simply stop playing the Belief with the complaint that it is never true. If the player knows, and then makes a choice as the PC, then he’s in on the hijinks. It becomes a bit meta, yet the player gets to play along with both player knowledge and PC knowledge. I assure you the transparency will make for a better experience in this case.