I like the idea of a turn check/raise (probably check/shove given stacks). I sometimes check flop with the intention of raising as well. Fishy players bet any piece and call a raise there, plus they also bet some bluffs because they put you on AK.
That said, against this super passive player I think you might have a value bet on the river. I would make it about $5.50 and fold to a shove. I don't think he folds 2pair to a bet here and neither does he raise worse than Tx. The more showdown happy he is, the better a bet becomes on the river.