Maybe check it down after the Q hits on the turn? If you know he'll check, that's pretty nice for you, with an overcard out and a weak kicker. And you didn't have enough chips to worry about building the pot just in case you're good. What was in the pot already would've been a lot for you.

Against a normal player, checking the turn and river would be begging for a bluff, but this guy is extremely passive, so it may have worked.