Playing a bluffcatcher against a massive range advantage.

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Playing a bluffcatcher against a massive range advantage.

BN: $58.32 (Hero)
SB: $185.87
BB: $51.23
UTG: $56.93
HJ: $50
CO: $63.27
Preflop ($0.75) (6 Players)
Hero was dealt A J
UTG folds, HJ folds, CO folds, Hero raises to $1, SB raises to $5, BB folds, Hero calls $4
Flop ($10.75) A K 8 (2 Players)
SB bets $6, Hero calls $6
SB plays on the tighter side as far as 3betting from the SB goes so far. On the board I'd guess I'm ahead of 85% of his range before he bets the flop, but doing considerably worse once that happens.
Turn ($22.75) A K 8 5 (2 Players)
SB bets $12, Hero folds
Looking at my defending range I think I should defend considerably less than 1-A here on the flop and turn but it's unclear how much I should actually defend. Here's an example strategy:

FLOP: call AT+, all flush draws (includes more suited aces), all pairs with at least a bdfd, all gutshots with at least a bdfd.

TURN: call AQ+, all flush draws

RIVER: call AQ+

This would leave me folding 62% on the flop, 52% on the turn, and 47% on the river (more or less depending on whether my flushes hit or not), and a ton of his range would still have clear value for three streets.

So yeah, given that there's no feasible way to not bleed a billion bbs here what am I meant to do? I can't even really just fold infinity hands on the flop because I'm already folding all the hands on the flop which don't have reasonable odds to improve to something which can happily call the river.
Final Pot
SB wins $21.49

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