But there might another way to avoid bubbling which I have missed. Maybe the compressed air is under so much pressure that it becomes denser than the water?
Imagine an cavern shaped like the great pyramid of Giza, full of water. They dig two pipes, an air entry pipe to the point at the top, and a water exit pipe to one of the corners at the base. Air pumped in at the top pipe will almost evacuate the cavern, before the water level drops to the point where air could get to the exit pipe.
What this means is that depending on the shape of the cavern, they may not be able to utilise its whole volume. In the worst case, if its roof were entirely flat, they would not be able to use any. They can use the volume of a section, bounded by horizontal planes, from the air entry pipe either down as far as the exit pipe, or (if the roof dips down between the two) down as far as the lowest point on the highest possible path between the two.
That's assuming totally vertical pipes. I know they can curve them a bit, but I don't know if they can curve them enough to enter a cavern from below. If they could, then they can utilise the entire volume, as long as they can identify the lowest and highest points
