I have faced this so many times vs fish - i understand how you feel
You played fine
flop: I bet pot
turn:
usually i shove turn after betting pot
as played:
i bet small like 1/4 or 1/3 pot nowadays (if i am confused between checking and betting 1/2 pot - i bet something smaller in between)
river:
check is fine given the run out
SB checked twice - fish with straight or flush typically cant keep quiet when they hit - again you can bet small - 1/4 pot
I will be hesitant to bet/fold river with a set on this run out.. i will bet/call