It's hard for you to have any logical bluffs here since the board is a complete rainbow and fairly dry. I think you can actually just 1/3 pot the flop instead of 1/2 pot. You pick up quite a bit of equity on this particular turn card. I think you should barrel turn like you did. The K helps you equity wise and you can easily have many kings. Should be a scare card for him.

I threw this is PIO as I was typing it and was surprised to see it checking QJo on the river almost always on this card. It's betting Q's, J's, A's (obv), 9's (obv) and checking most everything else. I expected to see a 3rd barrel tbh. I am assuming that the presence of many busted draws 64, 86, 98, 43, etc makes it want to save money against his winners and possible win at showdown if he checks back river.

Hawks, it might be because a solver is going to balance by calling stronger hands on the turn rather than raising for value, so naturally ends up with a stronger range then most regs here would. Think I like this combo a lot to barrel and overbluffing shouldn't be too big of a problem vs most reg.

I like this line with a few thoughts:

1) 1/4- 1/3 pot as suggested by Hawks on flop looks like a better idea.
2) Personally I like overbetting this turn as I expect to have the equity edge on this turn card against BB flop flat range - would be interested if PIO would agree
3) JQ blocks JT, QT, KQ, KJ and unblocks all his missed GSSD - like the river bet

I like 3 barreling this hand. You have good blockers to villain's best top pair and middle pair hands and don't block any of his obvious double floating hands except for J9.