KK vs a 4-bet preflop

partypoker fastforward, Hold'em No Limit - $0.05/$0.10 - 6 players
Replay this hand on Upswing Poker

UTG: $7.84 (78 bb)
MP: $23.70 (237 bb)
CO: $13.08 (131 bb)
BU: $21.45 (215 bb)
SB (Hero): $11.93 (119 bb)
BB: $13.96 (140 bb)

Pre-Flop: ($0.15) Hero is SB with K♦ K♣
3 players fold, BTN raises to $0.22, Hero 3-bets to $1, 1 fold, BTN 4-bets to $2.55, Hero calls $1.55

Is there any value to flatting a 4-bet here instead of pushing all-in? Seems I am only folding post-flop if an ace comes out.

Flop: ($5.20) J♥ 9♥ 7♥ (2 players)
Hero checks, BTN checks

I think this is fine, we do not have a flush draw, so we are most likely dominated.

Turn: ($5.20) 2♦ (2 players)
Hero bets $3.70, BTN raises to $11.50, Hero calls $5.68 (all-in)

This should be a mistake - the only two hands we are getting value from are QQ and AK with flush draws (and we block AK a lot). Even if we decide to bet we should fold to a raise.