AK - polarizing bet or turning my hand into a bluff?

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AK - polarizing bet or turning my hand into a bluff?

Blinds: $0.05/$0.10 (6 Players) BN: $12.24
SB: $17.33 (Hero)
BB: $4.52
UTG: $38.19
MP: $27.05
CO: $26.22
vs random reg
Preflop ($0.15) Hero is SB with A K
2 folds, CO raises to $0.25, BN folds, Hero raises to $1.00, BB folds, CO calls $0.75
Flop ($2.10) A 7 2
Hero bets $1.08, CO calls $1.08
Turn ($4.26) A 7 2 8
Hero bets $2.19, CO calls $2.19
His range is bluffcatcher-heavy, and my hand is too strong for me to XC the T, I think, and his range is pretty capped here, so I'm going for 3 streets.
River ($8.64) A 7 2 8 2
Hero bets $13.06 and is all in, CO folds
Sooo, the R is obv an excellent card for me, and I'm facing a pretty capped range here. I didn't like a small bet, because I'm not vbetting 3 streets with a weak Ax, so I think I need to bet big and polarize my range. So I went for an overbet to make him perceive me as desperately trying to make him fold Ax.

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