Even though he is a calling station, I think shoving the river is just too strong to get value from the hands that we beat. He won't call a shove with a lower Q (after the K on the river) and with JJ and lower.

If we beat most pocket pairs and lower queens, we should bet an amount that would be called by these hands. Therefore, a gay bet of 1/2 or a bit lower is just fine if you know what to do when he raises (fold, obviously). If you're not prepared to fold to a raise, just check behind and make a note.

His low bets on the flop and the turn are blocking bets. I can see he checking to you on the river a hand like TJ that just made a straight or trying to play lower PPs cheaply. However, those low bets have different meanings depending on the players. Once a donkey was min betting a lot on the flop, so I raised him in the first opportunity and he came over the top. On the flip side of that, when you just call, you're screaming weakness and they can exploit that on subsequent streets.