Did you consider patting on the final draw after having bet?
The arguments against doing so are that your draw is very strong and somebody might be pat but passive behind you.
But, in situations where one of your opponents might be playing the wrong game, there is substantial extra equity to patting something that can beat ace-high or a busted flush draw.
Did you consider checking the third round? I would tend to bet, as well, to try to get it heads-up, but your edge may be modest or even negative in this situation.