5NL - Unsure how to continue vs checkraise with TPGK on paired rainbow

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5NL - Unsure how to continue vs checkraise with TPGK on paired rainbow

Blinds: $0.02/$0.05 (6 Players) MP: $5.00
CO: $6.75
BN: $5.59 (Hero)
SB: $4.07
BB: $9.72
UTG: $5.23
I have no notes on BB, I've only seen about 27 hands. Over the small sample he was 21/17 with an AG of 8
Preflop ($0.07) Hero is BN with 9 A
3 folds, Hero raises to $0.10, SB folds, BB calls $0.05
Flop ($0.22) A 4 4
BB checks, Hero bets $0.15, BB raises to $0.45

In the moment of play I couldn't figure out what sort of range might check/raise here and how to continue.

With limited information I assume he defends BB with (if assuming still about 20%):
22-TT
A2s+, K9s+, QTs+
A7o-AJo, KJ-KQo, QJo

and 3bets AA, KK, AK, AQ, QQ, JJ

With my flop bet I'd probably expect most pairs >= 4 and any of his aces to call, and raise with perhaps 77-TT or AhXh hands with the pair and a backdoor flush draw.

If I call, I'm ahead of his middling pairs and a lot of his aces, and with 80% equity vs his range with my exact hand if we assume like above that it is: TT, 88-77, 9h9s, 9h9c, 9s9c, AhJh, AhTh, Ah9h, Ah8h, Ah7h, Ah6h, Ah5h, Ah3h, Ah2h.
If I raise maybe he continues with TT-99, AhJh, AhTh which is about 36% of his raising range. So we could fold out a lot of hands and fold to a shove, or just be in a weird spot where we give up if he just calls.
So if I raise to $1 (is that 2/3 pot) and he continues with about 36% of his raising range I'm assuming the EV is something like:
(0.64 * 82) + (0.37 * 65)
52.48 - 24.05
+28.43c

Folding seems like bad since we're often ahead.

If I called, would we essentially want to check/call the turn and be ready to fold the river if the board gets scary and bets large?

Feedback welcome if more/different information required in my own analysis of this, I'm expecting there to be a fair chunk of mistakes in the assumptions/review. I'm as much looking to improve my review process as understand the hand.

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