[mntndrew]

A fair amount of the time, you have the best hand on the flop with this flop missing anyone who played high or medium cards. It would be very vulnerable without the additional draws, but those draws make your hand strong.

If you're behind to A2 or an overpair, you still have a lot of outs -- up to 11.5, with 8 straight outs, 2 4's and 1.5 for the BDFD. In the rare instances you're behind the nut straight, you still have chop outs in addition to the BDFD. Bet the draw on the flop.

Definitely continue on the turn without a club. You're getting 10:1 and have way more than 4 outs even before counting implied odds.

I think the river is a clear c/r unless you've noticed your villain is particularly passive and sees lots of monsters on the river. Your flush is well disguised, and it's hard to see a straight checking here. Even a set shouldn't have too much to be afraid of here, and you've already stated that an overpair to the flop wouldn't have bet the turn when the K hit.