I've always considered that systems that do not use heat as its prime mover
should not be bound by thermodynamics.
Some of the laws specify thermal processes. But the 1st law of thermodynamics is the law of conservation of energy, and applies to all energy forms. People have been searching hard for violations to this ever since it was discovered, but it's never failed yet.
But yea heat engines are pretty useless at converting energy. They're bound by the 2nd law which restricts their theoretical maximum efficiency to much less than 100% in most cases.
As a side note, compressing air is a thermal process. It generates heat, so it's less efficient than an elastic spring.
To make it sink you have to use energy to compress the air. If you recover that while it floats back up then you can't recover it from the expansion of the bottle/piston. So you come out even at best.
If you make it sink so deep that the pressure of the water compresses the air in the floater automatically and pushes it down further, then to bring it back up you have to expand that air. Doing that means lifting the column of water above it. So the deeper it goes the more energy you need to do a complete cycle.
