Villain is a 19/16 over 300 hands
AF 2.8, 3bet 6 %
Cbet 64 %
cbet turn 1/2

No Limit Hold'em $0.05/$0.10
PokerStars
6 players
Formatted by pokercopilot.com - Mac OS X hand history analysis and tracking

Stacks:
UTG - UTG ($10.71)
MP - MP ($10.15)
CO - CO ($3.84)
BTN - BTN ($10.00)
SB - SB ($9.90)
BB - Hero ($10.00)

Preflop: ($0.15, 6 players) Hero is BB with K A
1 fold, MP raises to $0.40, 2 folds, SB raises to $1.20, Hero calls $1.10, 1 fold

Flop: J 6 3($2.80, 2 players - SB: $8.70, Hero: $8.80)
SB bets $1.80, Hero calls $1.80

Turn: J ($6.40, 2 players - SB: $6.90, Hero: $7.00)
SB checks, Hero checks

River: 7 ($6.40, 2 players - SB: $6.90, Hero: $7.00)
SB bets $2.20, 1 fold, Uncalled bet of $2.20 returned to SB

I hate these spots! Should I 4 bet fold pre? Or should I dubbel barrell when he checks to me on the turn? I'm really struggling in these situations with AK..

Would he ever fold QQ+ if I bet turn and river?

Did you consider MP, and what kind of range would villain 3bet him with?

I call the river, otherwise the flop and turn play does not make sense. By calling the flop and checking the turn you play for the sd-value of your AK, and since the river does not change anything, your plan should still be to realize that sd-value.
In other words: if you plan to fold river you should play the flop or turn differently. But since i like the turn and river play, just call the river.

To your question: Villain will never fold QQ+ to turn and river bets. What should he be afraid of?

4bet pre surely? I dont like to flat because we're going to be oop for the rest of the hand vs MP (potentially). As played, SB has 63% cbet which means he's playing pretty straight forward on the flop, you missed so just fold.

Oh one last thing, you didnt bet flop, you just called so you wouldnt be barrelling if you bet the turn.

4-bet/fold

What do we accomplish by this? I dont think the SB will call the 4bet oop ever, so we practically turn AK into a bluff. It makes much more sense to call and play a pot IP vs a SB-range that contains many dominated hands.

Coldcalling this pre is pretty bad imo. Calling the flop bet is even worse.
Not exactly sure what you thought process is here.