Suppose Nocturnal, Molag Bal and others all had a claim to the LDB's soul after death.

Who's the most powerful out of this that can claim the soul for him/herself and basically tell the other princes to, basically GTFO?

The Dragonborn's soul is that of a dragon's -- I'd say Akatosh has the greatest claim to the soul, not any Daedric Prince.

What happens if you leave Akatosh and Shor out of the picture?

Hermaeus Mora.

Tell that to durny in dawnguard, does not look like akatosh is gonna save him any time soon.

Looks like to me the ideal masters own him and Tsun says that nocturnal owns your soul if you are a nightengale.

In case of our Zombie Dragon probably it applies only in SC, on Mundus however most likely Akatosh would claim him... or Db... or he will stay "dead" like other dragons... nevermind.

But what if he (lets say an Argonian so Soverngarde might be out) pledged his service to the princes and became a vampire/werewolf nightingale listener/ champion of the prince?

Nocturnal may call dibs on his soul but what if Hircine wants it too? Who's the most powerful of them all that the other princes will simply back off on his/her command?