I'm sure I don't understand the scale of this. I would however think that you can bring huge industrial pumps and do some plumbing to a suitable place to dump the water and turn them on. Holland once send 7 million ton a day of pumping capacity to England to help them out. Could that not make a serious impact on the water levels in the cave?
Industrial scale pumps certainly exist, but they aren't exactly small and easily portable. I'm not familiar with the one Holland sent England, but I suspect it was pre-assembled on a ship?
The highest capacity pumps in New Orleans can move roughly 10,000 gallons a second, but they require a supply "hose" that's ~8-10 feet in diameter to reach that volume. You'd need to get an enormous hose to a low point of the cave system, which may not be near the entrance.
Then there's the question of what to do with the water you pump out. Maybe there's a river nearby that can absorb it?
It is doable, but it's neither simple, not particularly quick.
Also, the pumps have to be at the lowest point, pushing the water up - a siphon has a maximum height that a vacuum can draw water up of around 10m.[0] You can't just stick the pump at the entrance and then shove a pipe hundreds of meters down the cave system, you need to take the pump down there and run the hose back to the outside.
