Since he checked back turn, I would doubt he has many J8 in his range. He almost never has KJ, either.

On this river, he can have a lot of Q- and K-high straights and I don't think he is folding the latter to our river lead. Although we can fold out Q-high straights with some frequency, I don't think we run into them + fold them out often enough for this bluff to be profitable. Instead, I think we will run into K-high straights reasonably often, which will just snap our bet.

If we check to him, he almost certainly should value bet the 2nd nuts, and if we were deeper, I would like x/r bluffing with our AA blocker for that reason. But here with SPR=2, I just can't see him bet/folding the 2nd nuts, so I'd just check and give up.

Interesting thing is that since he didn't barrel on turn, he has a lot of give ups and our hand actually has some amount of SDV against them, but we can't really call if they turn them into a bluff. So it might sound like a cool idea to bet with our AA and fold those bluffs, but again, I'm not so sure if we won't run into his range that he just snap calls us with..

Ok so I wrote my comment before reading midori's and we are actually very close in conclusions, which is nice.

When you 3-bet and check call the flop and then he only fires the flop and checks back the turn, the action is consistent with you never having KJ and him having a one pair+draw type hand.

The river completes some but not all draws:
Draws completed: Q8xx, J8xx, 98xx, KQxx, KTxx, K9xx
Draws not completed: QJxx, TJxx, J9xx

On the balance of it I would assume we could get him to fold Q high straights but not K high. I like betting this river as a bluff to get him to fold Q high straights. It also takes away his opportunity to turn his draws not completed into a bluff on a good scare card. Unless he reads it all correctly and re-bluffs you, lol.

Fwiw I would consider leading the turn since we pick up some equity and it sets us up nicely to bluff the river. We are usually only getting raised by KJ which represents such a narrow portion of what he can have.