In general, I think it’s important, for meta-game reasons, to bet hands like this on the river (see my article Value Flows Downstream). If you aren’t betting the second nut full house here, then it will be very difficult for you to take advantage of your position on boards like this. You either make it too easy for your opponent to call you down with any pair or even an Ace, or you pretty much never double- and triple-barrel and therefore don’t get paid off when you have the overfull.
The only problem here is the bet sizing. On the turn, I made an extra big bet because I wanted to extract the maximum from a K, even at the risk of narrowing his range to mostly trips. Having thus narrowed his range, I think I need to check back this river. Had I checked or made a smaller turn bet, though, this would definitely be a value bet on the river:
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Seat 0: Hero ($5996.50) – 
– 
Seat 1: Villain ($12961) – 
PRE-FLOP:
Hero posts small blind $25
Villain posts BIG blind $50
Dealt To: Hero
RAISE Hero ($150)
CALL Villain ($150)
FLOP:
Pot: $350
CHECK Villain
BET Hero ($200)
CALL Villain ($200)
TURN:
Pot: $750 
CHECK Villain
BET Hero ($650)
CALL Villain ($650)
RIVER:
Pot: $2050 
CHECK Villain
BET Hero ($1250)
CALL Villain ($1250)
SHOWDOWN:
Hero:
Villain:
Villain collected $4499.5 from main pot with full house, kings full of threes
SUMMARY:
Total pot: $4500 Rake: $.50
Final Board:
Seat 0: 5d 5s Kc Kd 5c: full house, fives full of kings. – Net Gain/Loss: ($-2250)
Seat 1: Ks Kc 3c Kd 3s: full house, kings full of threes. – Net Gain/Loss: ($2249.5)
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