Preflop
BTN starts the hand with less than 16BB, and with a 3BB raise he has already put almost 20% of his chips in the middle. In that situation you do not want to have any calling range from SB, so this is a mandatory push or fold situation. You are a little deep against BB with 27BB, but if you push, he has to be extremely tight calling it off due to the final table ICM. So for me this is a no-brainer push, but if you really think, BTN is only opening to 3BB with super premium hands, then you can nit-fold and let BB do, whatever he wants to do.
Flop
SPR (Stack to pot ratio) is only around 1,5, and you flop TPTK. This is even more clear than preflop, and the absolutely only play to ever be made here is to check-jam. What are you holdning back for? Are you scared, you are beat? Or that he wont call with worse? Just get it in man and be done with the hand.
Turn
Because you completely butchered both preflop and flop, now you are in this super awkward situation, where the opponent has less than a pot sized bet left, and because a bad turn card rolled off, you have no idea, if your hand is good or not. I dont know, what I would do here, but the much more important point is, that you should never have been here, because you should have ended the hand preflop or at least on the flop.