There could be a possible straight draw on the flop, holding 9,10 or a gutshot holding 6,7. With top pair, checking in a multiway pot on flop with possible straight draws can be dangerous. In multiway pots, drawing hands increase in value and pairs decrease. By checking the flop you allow drawing hands a chance to outdraw you with a free card.
Bet enough to give improper odds for a call to a straight draw hand, so even if he does call, he will be on the defensive on the turn when you bet again.
Obv its super standard to 4 bet prflp and OP knows that. Its also just fine to flat on occassion so I don't have any problem with Preflop as long as you (OP) think villain will make a lot more mistakes post flop than preflop and UNLESS you know that villain only squeezes for value. If the squeezer always has AK+ when he makes this play then NOT four betting is GOD AWFUL. So, as far as PrFlp, meh, whatever..depends on opponent(s).
Recapp; PREFLOP; I think 4 bet is standard but flat isn't terrible Flop; Bet something, start building a pot TURN; you missed the bet on flop, so you have to lead turn but I'd go bigger so I can get stacks in on the river against KK and other second best hands (you already under repped PF and on the flop) your bet might look like an all out bluff on turn I would bet pot sized for sure. He's never calling with no pair, he's never folding JX or better as long as you don't REALLY overbet. River: as played I think you should make a very small bet and hope to get looked up by some hands you beat (prolly bet about 1/3 pot) then you can fold to shove much easier. You prolly could have avoided that awkward SPR by thinking ahead when sizing your turn bet. You had to hate folding an over pair you under repped to a shove getting pretty big pot odds, no?
This is one problem with taking your slow plays too far. Villain NEVER puts you on AA here and could be shoving second best hands for value/bluffing. Often enough to warrant a call? Prolly not, but i think its closer than most prolly think. Whatever he has he did something strange at some point in the hand, how much crazier is it to think that he has KK, AJ, or AQ than other stuff that beats you? I fold as played, but hate it. And just barely.