High pressure systems alter the density of the water throughout the water column, and this will of course affect pressure at the seabed. This an indisputable, observable fact.

No, it is rubbish.
There is no significant change in density due to air pressure. The entire pressure of the whole atmosphere is equivalent to about 10m of water. Changes in pressure only centimetres.
Even a full 10m does not compress water!

Even if it did, air pressure changes would still not be transferred to the sea bed.
The actual effect of changes in air pressure is to raise or lower the sea level, which negates any pressure change at the sea bed, or at any given depth.