This is a spot I used to run into trouble. I personally would lead turn and some rivers because as it stands, it's pretty much cheaper to NOT c/r and occasionally bet river. If you think he could have hands like TT, 88, 77, etc, then betting river is a waste of money since he would probably fold turn in 99% of cases. If he flats, you're beat whether that means, the weird sets (like he has) or AK, which you said was a big part of his range. If you're worried he could float turn with draws, I just question what draws he could honestly have? AT? There are few and most anything that whiffs bets the flop. In summary, I like to lead turn and give up in many cases on the river. Oh also, river SPR is too small to have much of any fold equity. I think if he flats this sizing on the turn, he isn't folding much of anything and could call with 2nd pairs on river shove.

Just give up, honestly, you have better hands to bluff with. Villain should have both KK and QQ combos in his range
Yes, he is probably folding AK combos ott, but he also should cbet them fair bit/he might decide to station off for some reason.

I think that you have negligible fold equity otr.