First of all don't post results.

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Folding because you think he has a higher set is stupid.

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Not in this situation. About the only hands that cold call that flop overbet (don't do that by the way) are AA,KK,AT+ (with a bias towards AQ+ because you said he was tight passive). No other hands call this. Out of this very precise hand range, 2 hands have us drawing 22:1 and AQ has us drawing slim. AK/AT we are way ahead of. Turn check from a tight passive doesn't mean too much alone, because that's what passive players do- check. However, couple this with the flop cold call and the river overbet/push, and the turn looks extremely like a missed check-raise.

That said, he is shortstacked. I'm absolutely showing down bottom set for a pot sized river bet.

I would have bet the turn, and call his push because you are pot committed. You are losing value by checking this turn.