Poker Rivered a set with short stack - Value bet or Showdown?

SB stack: 32,000 chips
HJ stack: 56,000 chips
CO (Hero) stack: 23,000 chips

Blinds 400-800

*** PRE-FLOP ***
UTG+1: limps
HJ: limps
CO (Hero): raises 3,600 with :8c4:
BTN: folds
Small Blind: calls
Big Blind: folds
UTG & UTG+1 fold
HJ: calls
Pot Size: 12,400 chips

*** FLOP *** :ad4::10d4::9c4:
SB: checks
HJ: checks
Hero: checks
Pot Size: 12,400 chips

*** TURN *** :ad4::10d4::9c4:
SB: checks
HJ: checks
Hero: checks
Pot Size: 12,400 chips

*** RIVER *** :ad4: :10d4: :9c4:
SB: checks
HJ: checks
Hero: bets 6,000 (I bet to get paid by weak aces, tens, and possible two pairs not wanting to lead into a straight.)
SB: raises to 18,000
HJ: folds
Hero: calls (In my mind, there aren't enough hands he could have to fold to and player is capable of shoving two pair as a bluff even with the short stack sizes. He's been active and has been bluffed out of hands with air balls on the river. He might be fighting back as he just hero-called with bottom pair.)

Is the pre-flop size too small? Is shoving 28 BBs better?
When the turn checks through, is shoving, betting, or checking the best play?
When the river checks through, is taking the showdown or value-betting better? If the doesn't come then we definitely go to showdown.


SB: shows
Spoiler alert!Click to Reveal!

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Source.