I thought of that, too, but there is no need to be able to detect it from the other side of the earth. Timed depth charges and sensitive seismometers should be enough.
More ping pong balls also should work. The first few billion may get stuck along the ceiling of the cave, but if you drop enough, some will have to come out somewhere.
Only if the flow rate is high enough to defeat the buoyancy. Not guaranteed at all.
Also, there was no indication what the geography is within the hole. Organic material will eventually break down and waterlog enough to lose its buoyancy. Those ping pong balls and rumored car, though...
Well, cutting open is perhaps a bit drastic, but how about just stopping the river for a day or so? After all, the OP mentions that this hole appears at the split of a river, so diverting the water to the other arm of the river might not be that hard...
I've learned years ago that even the "interesting" things have little chance to happen if they cost money. While visiting one of the most interesting caves in Poland the guide said that geologists strongly suspect that there's an entire tunnel system underneath it, and have suspected it for 40-odd years. So I asked - well, it should be fairly easy to check with the right equipment, no? And he said that yeah, it should be fairly easy, and even though multiple universities expressed interest in doing it, no one is willing to cough up the money for it. I guess this is very similar - of course it would be extremely interesting to find out where that water is going,but redirecting the flow of water would be very costly, and apart from satiating some scientific curiosity there's little reason to undertake such large engineering project.
Just tell them you think there is a train full of Nazi gold in those tunnels :-). Something we don't have enough of these days are "gentlemen explorers" folks who are independently wealthy and spending their days exploring interesting places.
I had a similar experience here in a shallow cave in Texas. When you reach the 'end' you really just reach the point at which the cave is flooded. The cave has been well known for over a hundred years and likely goes on quite a ways. No one really has an interest in exploring it. It'd be an expensive and likely just discover a bunch of rock and water.
With respect to cave systems, it seems like that the key is finding a previously unknown species. Or find a species in the cave that was thought to have a very limited distribution.
I bet you could do something neat with small tethered submersibles that you remotely control. Of course, they might not be small enough, but at least it would be safer.
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.
...where would you put a day's worth of river water?
Random internet search shows that a small river flows at about 1000 cubic feet per second, which is about 30 cubic metres; that's three million cubic metres in a day, which is three million tonnes (water is so convenient). The biggest supertankers can carry about 300,000 cubic metres, so you'd need ten of them...
"...where would you put a day's worth of river water?"
The other side of the river, by diverting all the water down the falls. River flows certainly vary by more than half in this part of the country/world (I'm in Michigan but it's not so different), so if you do it during a down time of the year you would not be creating a flow the rest of the river bed could not handle.
I'm not saying this is a "good" idea but it isn't a bad one in the sense you're suggesting. There is a place to put the water.
