3 handed play , Villan is a weak tight player , has 4 bet only twice in 250 hands and decided to check down AK once when he missed.

SB : 800 $s(400bbs)
BB(Hero) : 412 $s(206bbs)
BTN: 200 $s (100bbs)

Preflop : Hero Hand : k♥ 2♥
Btn folds, SB raises from 1$ to 6$ , Hero makes it 20$ , SB makes it 61.5$ , Hero calls

FLOP (pot -123$) : 9 ♠ A ♠ J♥
SB checks , Hero bets 54$s ,SB calls

Turn ( pot - 231$s) A ♦
SB checks , Hero bets 121$s , SB calls

River (pot - 470$s ) J ♣
SB Checks , Hero (176$) ?

preflop once he 4bets his range is QQ,KK,AA or AK or may be JJ and 1010 sometimes and i guess he is never getting out of line with anything worse .
I called in position since i knew i could get away if he continues aggression or if the board runs out bad he might check and give up.

on the flop
when he checks , since AK is a big part of his 4 bet Range and i block 25% of AK and 50% of kings i thought he might have Queens
I decided to bet and continued the story.

on the river I'm left with 1/3 of the pot , he is pretty much priced in .and he might call me with QQ and KK since he is blocking most AK and AQ
should i check and give up or shove and hope he has QQ or KK and folds .

my perceived image is loose agg .