Small stakes tournament, 6 handed tables.
3rd level, blinds 100-200, 8,000 to 11,000 stacks at the table.
I'm UTG with T9d, call 200 SB and BB call.
Flop is 78dKc. Checked to me I bet 400 SB folds, BB calls
Turn Jh (I hit the straight and still have flush and straight-flush draw), BB checks I bet 900, he re-raises to 2,000.

At this point I'm way ahead. I think he could have AKd, KJd, just top pair or two pair.

I 3-bet shove for my remaining 7,000 he snap calls and flips over Kh,7c for 2 pair, when I flip my cards for the straight, he sighs and states I didn't see the straight.

Of course, the river is the Ks and he hits the full house and I'm out.

I can't help think that maybe I shouldn't have shoved, that if I just called the 2,000 on the turn, if he bets large on the River I fold with the 2 Kings on the board. Or maybe, I 3-bet to around 3,600 and if he shoves I fold to keep half my stack. I don't see how I would play it that way without knowing the outcome.

How do you guys think I should have played this?

You got your chips in on the turn as a massive favorite and that's the part of the hand you're questioning?

Don't limp utg, drawing hands are very difficult to play out of position. Raise or fold.

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He is right, plus when you raise UTG with suited connectors you are effectivly disquising your UTG open range, so you can get paid more when you hit, or when you have a monster preflop.

Also i wouldn´t put AK to big blinds range, cause almost everyone raises with this hand preflop.

You can´t be results oriented, you got your money in great (90% favourite against villain´s hand) so don´t beat yourself up about the outcome, next time villain won´t hit a miracle river card.

Ditto

I agree, I should have raised UTG with my hand which would have probably resulted in me taking the pot pre-flop.

Thanks for the feedback, I will use this information to work on my game.