400nl: Open ender in small 3-bet pot

Villain is 25/16/1.8 over 290 hands.
He seems to be somewhat fishy, certainly not a great player.

What to do on the turn? Check and take the free card or shove?

poker stars, $2/$4 NL Hold'em Cash Game, 6 Players
Hand History Converter by Stoxpoker

SB: $1,005.70 (251.4 bb)
BB: $106 (26.5 bb)
UTG: $436.10 (109 bb)
MP: $434.60 (108.7 bb)
CO: $416.65 (104.2 bb)
Hero (BTN): $477.50 (119.4 bb)

Pre-Flop: Hero is BTN with J 9
3 folds, Hero raises to $12, SB raises to $24, BB folds, Hero calls $12

Flop: ($52) T 4 8 (2 players)
SB bets $36, Hero raises to $146, SB calls $110

Turn: ($344) K (2 players)
SB checks, Hero ?

I think it depends on the dynamic that you two have had preflop. Ship it in? You only have $300 left. This would put JJ and QQ in a tough spot but if you guys have been 3 betting and 4 betting a lot his range is much wider

There wasn't much of a dynamic, we had only been at the same table for about 60 hands. He had not 3-bet me up to that point.

Did you see how villain plays flush draws on flop in previous hands? You line looks a lot like a draw, and if you push the turn, it looks like a flush draw that got there, so I don't think he can call unless he has a made flush himself, AdKx or a set. If you can rule out the made flushes (because he would have shoved flop with it or because he would check-call flop instead), then it's a pretty narrow range and pushing is fine. If you can't rule them out, there's a lot more hands that can call you and you have a much worse equity when called.

The Kd is like the most perfect scare card in the deck, but what is villain calling a huge flop 3 bet with? It is such a perfect spot to shove (and your flush draw might even be live) but seeing as villain didn't go away on the flop i'm not really sure how much fold equity we really have here. Folds out JJ/QQ and the wider part of his 3 betting range but yea idk it is pretty hard to know what villain's range is here without more reads than vpip/pfr/af