I'm far from being even slightly knowledgeable in this topic, but would it be possible to build this in very deep waters? Like a massive column containing a tunnel? Seems cheaper than digging a 1km hole.
You'd be anchoring a buoyant tube to the ocean floor, trapping all that energy. Instead, float a barge in deep enough water, drop a weight to generate power, raise it to store.
At the base of a 1000m shaft you'd have approximately 100 atmospheres of pressure. Not saying it cannot be done, but construction and materials cost would still be very high. Maintenance too. Plus you've lost the advantage of locating it anywhere.