FLOP: b25-35 on flop; because board is so dry there is nothing to protect against, so you are betting simply for value against Qx. The less you bet, the liter you encourage him to float.
TURN: Bet 1/2-2/3 pot. This is a straight value bet. First off, I don't expect the Vil to enter the pot HU particularly lite when presumably the "whale" has already folded in front of him pre flop (especially given his perceived image of you). So backdoor clubs hardly is a concern when the Q
is on the flop. Do you really believe this Vil is peeling you're 4/5pot cbet with T9cc, JTcc, 56s? So there is very little to protect against. You are betting here with the intention of extracting value from Qx, and possibly weaker depending on sizing.
As played, raising after he bets when we check to him would be grossly overplaying our hand. We are only getting called by better: 87, possibly 44; and getting 3b by sets+.
River: If you had bet flop+turn, the river is another value bet 1/2-2/3 pot.
As played, I also prefer betting 1/2-2/3 pot. Because we bet 4/5pot on flop and got a call, I would skew his range towards value hands. Because Qx checks behind too often and your hand is NOT strong enough for a k/r, betting for value is the most +ev line. If you had bet smaller on the flop AND the Vil was capable of floating lite, I would prefer you k/c river after k/c'ing turn.
Miscellaneous: You said Vil is competent and positionally aware. His perceived image of you is that you are nitty. You are playing 140bb effective. How deep is the "whale" at this point? We can assume if the "whale" is deep, the Vil will be less inclined to run a move on a nitty player. Why would he go after you when he is playing deep against the "whale?" Why risk the stack? I also assume (from relative positions and based on the fact you mention you have 3b a couple of iso-raises) that the Vil has already seen the "whale" fold before he calls your UTG raise in a scenario that is likely to go HU. I think given these circumstances we can skew the Vil's range towards strong value hands, like AQ-QJs, PPs. It is possible he just peels with TT or JJ and folds on the turn, but I think the majority of the time the Vil specifically has a strong Qx here. If you size it right, you can extract 3 streets from Qx in a situation where the Vil is incredibly unlikely to spazz out against you. By extension of this logic, this is a b/f on every street. Moreover, if PPs are a good part of his range, then when we check turn to him and he bets, we LOSE information. Had we bet and he raised, we could have confidently folded. When we k/c turn and then subsequently lead river for value and he raises, we don't know if he is raising because he flopped a set and we denied him the delayed turn raise, or if he is overplaying his hand because we have under-represented ours. As played (k/c turn + lead river), I would fold if he raised my river donk bet.
In short, I think you lose value with bet->k/c->bet line. Your hand looks like what it pretty much is AQ+.