I'll try to tackle this with some math, just to amuse myself.

I had originally assumed that ft.lb was a unit of power, but a quick google shows that it is actually an energy, with ft.lb/s being the power unit. The buoyant force of each bucket will marginally decrease as it rises, as a function of the surrounding water density, but this can be effectively disregarded as trivial.

Be that as it may, let us assume that each bucket can provide a total of 100ft.lb of energy as it rises from the bottom of the cycle. Over each 'interval' of machine time, this being the time between each bucket passing a set point, you can only extract 100ft.lb of energy, this being the difference in potential buoyant force given by raising N buckets 1/N of the way from bottom to top. Let us assume that one bucket passes the top each second, your total extractable power will thus be 100ft.lb/s, and not one erg more. Thus is exposed your first mathematical error, that you can extract the full stored energy of the system over and over again.

You will find, should you calculate the power and energy needed to compress and force enough air to fill a bucket from sea level to the bottom of the machine, that it will be greater than 100ft.lb and will thus require more than 100ft.lb/s to fill a bucket as it passes the bottom. And the bucket MUST be fully filled at the lowest point, any delay in filling only reduces the total energy that can be provided by it then rising to the top, in direct ratio to the delay in filling with air.

Needless to say, all this assumes that your machine has no bearing losses, drag losses through the water, power extraction losses, compressor losses, heat loss, leaks, etc, etc, etc...

So, no, you cannot fill buckets under water with air and let them pull your machine along, the idea is even less feasible than pumping water up to buckets which then fall down and pull a chain to provide power. The latter at least can be used to store energy, though a hydro station using one bucket of water per second would be rather on the small side (excepting certain very large definitions of 'one bucket').