K4s - A 4-bet bluff gets flatted IP. Now what?

[Carroters]

Villain is 18/15 over 280 hands. 3-bets 7% (10%) in the BB and has folded to 2/2 4-bets.

Think this is a pretty good spot to 4-bet. I have no idea what his flatting range is here, but Iassume it's pretty much just parts of his 3-bet value range since he obviously does fold to 4-bets and we have no real dynamic. First time I've 4-bet him this session.

So I either wanna 1/2 pot flop and jam turn to fold out JJ-KK type stuff, or just give up. I don't expect to see AK too often, AQ I guess is possible. Think I like giving up and think firing one is our worst option here.

Thoughts?

Note I make this roughly 11 preflop, converter being strange.

Party Poker No-Limit Hold'em, $0.50 BB (6 handed) - Party-Poker Converter Tool from FlopTurnRiver.com

BB ($77.90)
UTG ($15.70)
MP ($20.75)
CO ($42.65)
Hero (Button) ($53.10)
SB ($65.40)

Preflop: Hero is Button with K, 3
3 folds, Hero bets $1.50, 1 fold, BB raises $4.50, Hero raises $9.90, BB calls $6.40

Flop: ($23.05) 6, A, 5 (2 players)
BB checks, Hero?