Good call on the reserve air! Sperm whales need it for both detection as well as stunning their prey. The latter is still a bit of conjecture, but given the absolutely huge accoustic lense they have in that forehead of theirs...
On decompression: air goes in at 1 bar and gets pressurized by the surrounding waterpressure acting on the whale's body as the animal dives, but as a consequence the volume goes down, so: there should really be no decompression related overfilling (causing rupture) or caisson type bubble formation in the animal as it surfaces. You can't have oversaturation if no hyperbaric air entered the lungs to equalize the pressure at depth.
The strandings hypothesis based on (underwater) testing of new types of extremely loud sonar, is more likely related to internal damage and haemorraghing of the ears, mainly, not decompression. This type of damage was seen in strandings just following sonar testing, nobody seems to think that's a coincidence.
I know that there are reports on stranded beaked whales, showing signs of the bends following sonar excercises. How to explain that from a physicist's point of view, I wonder.... Partial gas pressure increases, forcing excess nitrogen into the bloodstream?
I also wonder whether the "recent" Spanish strandings of five Architeuthis in one week were likewise correlated to testing at all, hmmmm. Could just have been mating season, of course...