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I work with this stuff. It's not necessarily unfeasible given a sufficient quantity of properly sized pumps. The topography means fairly minimal lift is required so the pumps could run close to maximum rated capacity. There's a bit more to it than just calculating the volume of the tunnels; ground porosity means a lot of that water will be absorbed by surrounding soil. The opposite problem is encountered during dewatering - quite a bit of pump capacity is given over to pumping out voids, fractures, and porous ground so the discharge rate is several times the dewatering rate of the target opening.
The comments about rain are nonsensical. There's a serious difference between short-duration rain events, buffered by infiltration time, and a sustained pumping campaign. Same for the pumps "not holding out" and "becoming useless". Maintain them properly and have spares on standby. It's not rocket surgery, we do this stuff industrially 24/7/365.
It reads like a lack of equipment availability or cash to rent/purchase suitable pumps.