I play it the same.

The bet on fourth should be absolutely automatic with any fully live flush draw here. You have way more than a 1 in 6 chance of winning this hand, so you want money in the pot, especially at 1/2 when folding is not a commonly used strategy.

I don't really consider folding fifth or sixth here - there's little reason to suspect trip kings. I think that going for overcalls on the river is better than raising here.