So I 3bet and get coldcalled by button, who is running at 13/0 over 7 hands, so basically an unknown. (there is a load of 15/10 guys so I think we can assume he was one of those at this point)

We cbet and get called. The options on turn I went through were either:
a)bet like 1/2 the pot, fold to raise, and then bet again on river, making this quite a big pot.
b)check and call up to 1/2 psb and same on river, or fold if he bets bigger like he did

Now that he bet so big, I figure I am splitting pot at the best with AK, or he just improved his QJ or hit a set on flop/turn.

CRAI on turn did cross my mind but I didnt see what hand that I could beat will call, AK was really only hand that I wouldnt lose to.

How could I play this less terrible?

Grabbed by Holdem Manager
NL Holdem $0.10(BB) Replayer
SB ($6.98)
BB ($10.30)
UTG ($5.10)
UTG+1 ($10.87)
UTG+2 ($14.37)
MP1 ($15.59)
MP2 ($3.30)
Hero ($11.07)
BTN ($14.47)

Dealt to Hero K A

fold, fold, UTG+2 raises to $0.20, fold, fold, Hero raises to $0.65, BTN calls $0.65, fold, fold, UTG+2 calls $0.45

FLOP ($2.10) K 5 T

UTG+2 checks, Hero bets $1.35, BTN calls $1.35, UTG+2 folds

TURN ($4.80) K 5 T 9

Hero checks, BTN bets $4.80, Hero folds

BTN wins $4.48

Firstly i would raise bigger preflop, probably .80-.90, you could even go as far as $1 imo. .65 is too small. On the flop, your bet size looks pretty weak. On this board, i'm firing $1.50-1.65 if the pot is 2.10. Obviously this will be a bit more if your raise preflop is bigger. Turn is a bit tricky given how this hand has been played, another barrel from you will probably be committing and it's hard to define villains range due to his call of a smallish 3bet. If the 3bet had been larger preflop, it would be a bit easier to define his range.

I don't hate the c/fold. You could very well be splitting with AK, and the only hands you beat here are Kx and incomplete draws (not many). I think it may well be a good fold. In future though, don't let preflop min raisers get what they want by 3betting too small. I raise the same or near the same as my normal 3bet against min raisers.