check back small set on the river

PokerStars - $0.02 NL FAST (6 max) - Holdem - 6 players
Hand converted by PokerTracker 4

BTN: 112.5 BB (VPIP: 9.52, PFR: 9.52, 3Bet Preflop: 0.00, Hands: 67)
SB: 161.5 BB (VPIP: 100.00, PFR: 100.00, 3Bet Preflop: -, Hands: 1)
BB: 101.5 BB (VPIP: 33.33, PFR: 33.33, 3Bet Preflop: 33.33, Hands: 9)
UTG: 102 BB (VPIP: 50.00, PFR: 50.00, 3Bet Preflop: 0.00, Hands: 2)
MP: 160.5 BB
Hero (CO): 228 BB

SB posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has 4♠ 4♣

UTG raises to 3 BB, MP calls 3 BB, Hero calls 3 BB, fold, fold, fold

Flop: (10.5 BB, 3 players) Q♣ 4♥ J♠
UTG bets 5 BB, fold, Hero raises to 13 BB, UTG calls 8 BB

Turn: (36.5 BB, 2 players) 9♦
UTG checks, Hero bets 31.5 BB, UTG calls 31.5 BB

River: (99.5 BB, 2 players) 7♠
UTG checks, Hero checks

Here we flop a set. I try not to slow play it since it's a small set and the board could get ugly.

I raise villain's cbet on the flop and he calls, then he check/call the turn and checks on the river. I was afraid of getting raised by a straight or a better set, so I check back. Only later I realized than villain only had half of the pot in his stack, so I should have bet and put villain all-in.

What if villain had more chips left in his stack? Should I bet anyway and fold to a raise? Would anyone call a raise on that spot?