You say that he's LAG and not stupid. If he's not stupid, I don't just take this to mean that he doesn't cold-call with complete junk. I also take it to mean that he probably knows that you're tight and that he can possibly either use the free card play against you or just try to bluff-raise the preflop raiser off something like KQ, KJ, or some middling pair like 88-TT.

Therefore I think villain's hand is really undefined by the check-raise. He could be bluffing, semi-bluffing, slowplaying on the flop, or protecting two pair against a BB that he thinks will call one bet but not two. You have TPTK heads-up, and that's extremely strong against any kind of loose player, aggressive or passive. My instinct here is not to let him away with a semi-bluff and 3-bet the flop, planning to bet or raise the turn, and re-evaluate if check-raised or 3-bet on the turn. If villain caps the flop, I probably just call down.

Now, given that villain checks turn and river, I say he's on some kind of mid pair, probably 44 or 55.