If he is unwilling to act according to the lease which he put forth at the beginning of your arrangement--the agreement set forth to mediate your relationship--then he is NOT a man of his word and he is not to be trusted. Seeing as we don't live in ancapistan, we don't have any mediatorial body besides the state (for the most part, with a few exceptions, but here there is no exception). Unfortunate, but true.
As I see it, you should do something along the lines of this: Be of good faith with him and do not mistreat him, after all, you have a lease mediating your relationship with him. Besides, economics is not a zero-sum game, so the best ends is something which is positive for both of you. Help him to realize this. The additional information you gave, though, is a strong playing hand in your favor.
From my perspective, your words should go something like this (make certain your words to him can't be construed as blackmail--an even worse outcome for BOTH parties). Seeing as you don't believe in the state you would naturally be inclined not to report the infraction to the state. However, if you were to take and press the issue in civil court, the issue would likely be found out. Obviously you wouldn't because the legal costs would be too high--but, maybe, as far as he is concerned, your "uncle" is a "lawyer", though so it would be "mostly free". You realize this would be a bad thing for him, and probably the taxes would cost much more than the money you would owe him (assuming yearly rent is in the vicinity of 20-24k [yikes!]). I'm guessing you're in SF or NY, so additionally, you know that he will not have a hard time filling the house with new tenants. Therefore, it's in his best interest to keep to his word which was agreed upon instead of trying to cheat you.
Make certain, though, in whatever you say, that you aren't threatening him--just that the outcome of cheating you will be far worse for him. Be reasonable, and hopefully he will act rationally.