OK I will solve that AKQ game first, then I will try to show why I think the situation is similar in your scenario for this hand. Then you can prove me wrong.
Hero is IP with a K, villain is OOP with an A or a Q. Pot size is 1, bet size (all-in) is b. Villain can check or open shove b. Hero always checks behind when checked to and can either call b or fold when shoved into. It makes sense for Hero to always check behind when checked to, because Villain always calls with an A and always folds a Q, so there is no sense in betting. Now this game may seem unfair because Hero always has a K and Villain knows it. In real poker that would correspond to a Villain who is good at putting Hero on a range and the action on previous streets tells Villain that Hero has neither a nutty type of hand, nor complete air. Hero is not bad himself and he knows that Villain knows that, and he also figures that the range of Villain when he bets is made up of nut hands and bluffs.
Villain's strategy: always bet an A and bet a Q a fraction x of the time
Hero's strategy: fold a fraction y of the time
Preliminary remark: since villain bets all his A and a fraction x of his Q, it means that, whenever villain bets, he holds an A a fraction 1/(1+x) of the time and he holds a Q a fraction x/(1+x) of the time.
If both players play unexploitably ("at balance"), Villain's EV of betting a Q is zero (betting or checking a Q has the same expectation for him), and Hero's EV of calling is zero (calling or folding has the same expectation for him).
Villain's EV of betting a Q:
EV = y*1 - (1-y)*b
EV = y - b + yb
EV = y*(1+b) - b
Hero's EV of calling:
EV = (x/(1+x))*(1+b) - (1/(1+x))*b
EV = (1/(1+x)) * ( x*(1+b) - b )
When we zero both EV's, we find:
x = b / (1+b)
y = b / (1+b)
Which is the familiar bet / (pot+bet) ratio.
So now with your scenario for the hand above, the pot OTR after your $6.75 turn bet is $16.84 and the bet size OTR is $12.66. That gives us a bet/pot ratio b of 0.7518.
To be unexploitable, villain should bluff x=42.9% of the time with the bottom part of his range and check the rest of that bottom part. It also means that when Villain plays unexploitably and he bets, he has almost exactly 30% bluffs and 70% value hands in his range (a ratio of 0.429:1, just like in Spoon's signature's post).
To be unexploitable, Hero should fold y=42.9% of the time, and so he should call a seemingly massive 57.1% of the time (that's where I completely messed up when I said above that Hero should only call 12.9%).
As an aside, it intuitively seems like an awful lot of calling when we are crushed 70% of the time, but this is exactly made up for by the 30% times we win his bet AND all the dead money in the pot. Some reasons it may feel weird to call so much is 1) we are used to try and make fat +EV plays rather than 0 or near 0 EV plays (especially at microstakes), which is another way of saying that we tend to try to play closer to an optimal exploitative strategy than close to balance and 2) we are not used to play against Villains who have very balanced ranges themselves.
Before I go further and try to show why I think this game is a decent model of the river situation you describe with this hand, please let me know if the above is correct.
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