$5 NLHE 6-max: Missing value on river?

[GreenDaddy1]

Reviewing this hand, figured I missed a jam here on the river for value? Only 4 hands on villain. His post flop play is really not screaming strength to me. When the ace comes on the river I figure I am ahead (or at worst chopping) a very large % of the time.

As I played I think I let his large 3 bet size get into my head, and due to that just called off on the river. Going back over it though, I really feel villain narrowed their range through their post flop actions to hands I beat or chop with once the ace hits the river?

Pacific Poker - $0.05 NL FAST (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

Hero (BTN): 110.8 BB
SB: 55.6 BB
BB: 94.6 BB
UTG: 216.8 BB
MP: 40 BB
CO: 105 BB

SB posts SB 0.4 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.4 BB) Hero has K:heart: A:heart:

fold, fold, fold, Hero raises to 3 BB, SB calls 2.6 BB, BB raises to 13.8 BB, Hero calls 10.8 BB, fold

Flop: (30.6 BB, 2 players) 7:club: 3:heart: 6:diamond:
BB checks, Hero checks

Turn: (30.6 BB, 2 players) 3:spade:
BB bets 12.4 BB, Hero calls 12.4 BB

River: (55.4 BB, 2 players) A:diamond:
BB bets 22.2 BB, Hero calls 22.2 BB

Spoiler

BB shows Q:club: A:club: (Two Pair, Aces and Threes)
(Pre 29%, Flop 18%, Turn 14%)
Hero shows K:heart: A:heart: (Two Pair, Aces and Threes)
(Pre 71%, Flop 82%, Turn 86%)
Hero wins 94 BB

 

[vsawake01]

Two questions (and you know answers I think):1. Should I push? Yes.
2. Who will pay? AsQs more likely will pay.

Does the 3-bet range contain better hands?