Interesting spot, where I feel like not bluffing a river is a mistake.
Few questions.

1) What range are we checking, and then betting the river with?
2) Do we pot control Ax TP on this turn card?
3) What range do we want to value bet on this river? Thin value bet with Ax, and what sizing???

Personally I think that we can check AA, KK, weak 8x (87s 86s) and maybe the strongest flushes (QcJc QcTc) on the turn.
I think we should not keep betting Ax TP once the flush hits, but BxB would make sense. And this would be one of the bluffs we can have to balance the hands mentioned above. But what sizing?
Big for nutted hands, and ½ pot for TP?

2 NL (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

MP: 54.94 BB (VPIP: 48.94, PFR: 29.79, 3Bet Preflop: 9.09, Hands: 47)
CO: 115.68 BB (VPIP: 24.55, PFR: 18.09, 3Bet Preflop: 9.26, Hands: 395)
BTN: 107.48 BB (VPIP: 25.46, PFR: 20.57, 3Bet Preflop: 7.46, Hands: 504)
Hero (SB): 322 BB
BB: 334.93 BB (VPIP: 26.00, PFR: 19.59, 3Bet Preflop: 9.50, Hands: 7,197)
UTG: 195.47 BB (VPIP: 26.44, PFR: 17.24, 3Bet Preflop: 2.83, Hands: 272)

Hero posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has T J

fold, fold, fold, BTN raises to 2.5 BB, Hero raises to 10 BB, fold, BTN calls 7.5 BB

Flop: (21 BB, 2 players) A 8 K
Hero bets 7 BB, BTN calls 7 BB

Turn: (35 BB, 2 players) 8
Hero checks, BTN checks

River: (35 BB, 2 players) 9
Hero ???

Cbet bigger otf. I'm assuming you'd be checking most of your medium/weak aces ott. If you balance your turn checking range with nutted hands then you can credibly represent a hand our. Ofc this depends on how villain perceives you.

Not necessarily a mistake. I'd find a lot of people continuuing to check turn with AX rather than betting and always calling river.

These spots also confuse me, partly because it's hard to build an overbet range with an absence of relevant blockers (we usually don't 3bet any Q high offsuit hands pre so we never have a club blocker here).

I think in chop pot situation you want to proceed with a small vbet size that should force him to call some Kx to avoid overfolding. But then you don't get to bluff much.

Very narrow OB range of KK+99+8x. Bluff some QJs that has blocks his autocalls (AQ/AJ). You prob need an rng to regulate that effectively, or I guess you can do it with QJdd.

It kinda all depends what happens ott, is hero checking range? Is V checking back all his made hands <trips? Some guys will use the small size ott with a lot of their range/Ax at least in which case you get to vbet river with Ax more aggressively/larger.

What is villain's reaction to 3bets pre-flop?

Eyeballing it, really don't mind check/folding the flop, but c-betting same flop with 1 diamond (maybe depends on villain's tendency vs. 3bets pre-flop).

Thinking about turn/river.

This exact turn card seems to me we should bet with impunity because I think it caps the crap out of villain's range. Qc/Jc/Tc would be a close second to block more nuts but a couple combos of nuts is all villain has and then villain will sigh fold a bunch of rivers because no one bluffs that run out.

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I think this combo should go in our half pot range. We can balance with AT-AQ, 8x. I find it hard to find good bluffs to balance our OB range. We can select some combos that block boats like our weakest Kx and T9s.

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So you bet 1/3rd pot w/ AA/KK on the flop, check turn and over bet river?

mmmm... seems like wishful thinking a bit. I wouldn't bluff this river. I would expect any competent reg to call you w/ any Ax and Kx, even to most over bets. I'm not really sure there's a bet sizing that would get a fold. 1/2 pot river is more consistent w/ Ax of course, but what are you folding out?

I think if you're bluffing here you are thinking 3 streets from the start, otherwise I think you're getting looked up too often.