Villain is on the tigher side, at least on the button. He's playing multiple tables and I don't really have any reads so far. He's 3bing 22%. Everything's quite clear until the river.

Are you vbetting here and if yes what would be your sizing?

I think I can shove to get calls from his bluffcatchers and I think he would have shoved the river himself if he had something which got me beat. Right?

PokerStars - $0.50 NL FAST (2 max) - Holdem - 2 players
Hand converted by PokerTracker 4

Hero (SB): 204.06 BB
BB: 144.14 BB

Hero posts SB 0.5 BB, BB posts BB 1 BB

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

Hero raises to 2.5 BB, BB raises to 8 BB, Hero calls 5.5 BB

Flop: (16 BB, 2 players) 8 7 3
BB bets 12.98 BB, Hero calls 12.98 BB

Turn: (41.96 BB, 2 players) T
BB bets 27.3 BB, Hero calls 27.3 BB

River: (96.56 BB, 2 players) 2
BB checks, Hero ???

Don't think you can, no. In this spot, villain most likely has "8x or worse", or you are beat. And by shipping, which is the natural betsize, he simply won't call you with worse enough.

Flop is razor thin between three options imo.

It can go either way with reads. If villain is a high % turn cbettor in this spot I would shove. He'll have some 7x, 99 and all kinds of KT, QT, JT, 9T combos which look just like this when he shows up with them. Has he thin value bet? Has he declined value bets OOP?

I think OP's last question is very interesting. It revolves around the fact that going any distance up our list of hands we start encountering high showdown equity vs villain's checking range because the river is a brick. It seems to me that villain would want some kind of reinforced check calling range but also with such a low SPR (capped, who cares?), with us so heavily disincentivized to bluff with the bottom part of our range, and with villain still expecting many calls from worse by barreling a hand like AA, I am now not sure that he shouldn't be shoving every hand that beats us. And if that's the case or anywhere near it, then shoving would be optimal regardless of other reads.

Oh wow, I missread this hand completely. Not sure why. Thought villain x/called twice. Disregard my first comment. I think you should shove here.

i jam and play it the same

Yeah I did that. Thanks for your responses guys