NL10z Constructing our checking range after 3-bet on Qxx two-tone

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NL10z Constructing our checking range after 3-bet on Qxx two-tone

Blinds: $0.05/$0.10 (6 Players) BN: $4.00
SB: $10.05
BB: $10.05 (Hero)
UTG: $15.68
MP: $11.71
CO: $32.43
Preflop ($0.15) Hero is BB with A A
2 folds, CO raises to $0.25, 2 folds, Hero raises to $1.10, CO calls $0.85
Flop ($2.25) Q 7 8
Hero

Similar spot as the hand I just posted. A two-tone Q-high board which provides quite some pair and straight draw potential for the PFC's range which is composed of a large chunk of middling cards (J-7) warrants selective c-betting, but I don't classify this is a range check.

Basically I am wondering how we construct our checking range here. If we bet all top pairs and overpairs, then how to defend vs. multi-street stabs once we check?

  1. check stable overpairs (AA) but bet KK?
  2. bet TPTK but check weaker top pairs (QJs, QTs)?
  3. check nut flush draws with SDV (gives us flush coverage in the branch of game tree where we checked a heart comes on the turn) and bet weaker flush draws?

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