The water is just going to stop at the level of the lake, so it will still be flooded. By far the most likely answer is that it gets filtered out into the bottom of the nearby lakebed. I'm guessing there are pockets above the passage that the water flows through where floating debris gets trapped (full of ping pong balls, etc...) and the flow rate is too high for dye to remain detectable.

Damming off half of the river and then cave diving it would be awesome, but tremendously expensive and only good for answering a trivia question.