HU J7s, gutshot OTT - Bet or Check

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HU J7s, gutshot OTT - Bet or Check

Blinds: $0.25/$0.50 (2 Players) BB: $50.20 (Hero)
SB: $51.50
Preflop ($0.75) Hero is BB with 7 J
Hero raises to $1.00, SB raises to $4.00, Hero calls $3.00
J7s is well within my defending range here. So I call.
Flop ($8.00) A 4 T
SB bets $4.80, Hero calls $4.80
Given the backdoor equity, I think this is within my floating range too.
Turn ($17.60) A 4 T 8
SB checks, Hero bets $8.25, SB raises to $42.70 and is all in, Hero folds
Now I am a little confused. How do I construct my:

Bet/Fold Range (Weakest Draws without showdown and AIR)
Bet/Call (Strongest Draws and Sets/TwoP+)
Check Back (Pair + Weak Draws, TP, Second Pair and 8x)

Does this sound fair?
Final Pot SB wins $33.60
Rake is $0.50

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