Interesting theory - but wouldn't there have to be some gas involved (bubbles, cavities) as water isn't impacted by pressure (at least not at a level experienced while scuba diving)? The pressure though will certainly impact the (pressure) sensor in the bladder signalling that it is full. But my impression (no pun intended) is - as you also suggest - that the bladder actually fills up faster due to some physical / physiological effects while diving. Even at the surface at some point the lower part of the belly is palpably tense. My hunch would be that something speeds up reverse osmosis (mineral/colloidal concentration of two regions separated by a membrane evening out by water moving into the region with higher concentration); as I'd assume that the already present urine has much higher concentration in salts and proteins compared to the surrounding tissue.
I just had the idea that it might be due to water entering the body and blood stream through the skin (by means of reverse osmosis). The kidneys then excrete the excess water into the bladder.