I think checking back the turn is prob the right play though conservative. Opponents who are paying attention will no you never have a nut hand in your range, but they still have a few. I know it may seem like checking back is bad here but:

a) it's a 3Bet pot, so a lot of villain's range hits this board
b) you are drawing to a nut hand IP, but right now you have a relatively weak hand
c) you have to fold to c/r's

typically people don't mess around on the turn so as played its a fold, which is why I advocate checking back.