I think the first hand that you sucked out on was odd, then again you didn't say what your BB's was at this point as you said on the next hand that shoving with K,6os was really weak yet you shoved the K3 the hand beforee.
Anyway, as for the hand - as you played I don't think you could of really folded it in the end as you already put money in and was quite a big percentage of your stack.
I think betting is really bad on the flop because you are quite pot committed now so any scare card comes you're now stuck so either he was going to be shoving on you or folding so my approach would of been either shoving preflop, or shoving post flop. He most likely did see your bet as weak at this point so came over the top in the hope you would fold. If you did shove the flop though I can't see him calling most of the time..at this point and you do have fold equity left. If he just flatted(which he did) you have no lost your fold equity and you would be thinking what he has and now the problem he would always be calling the turn/river.
So yeah, the only thing I can think of here is shove pre(if you intended to bet the flop if a 6 came off) or post flop is the only play available. Betting is kind of pointless as you are putting in a big percentage of your stack and losing fold equity on the turn/river. The only time I'd bet like this is to induce a shove from him, so basically if I hit the King or two pair in the hope he shoves on you, other than that - any low pair like you did is a shove for me. You must remember, if he just flat calls here you are in a difficult situation on the and the river as both are getting away from the pot now. At least with shoving, he has something to think about, you lose all your fold equity from a flat call onwards.
Good job on final table though! Was a good call though, I would of made the call if in the same position as you and you got unlucky in the end, I just think it was shove/fold though at this point.