This is helluva question..I'v thought about this a lot lol. The way things are going I think the airforce has "Jurisdiction" over beyond atmosphere flying aircraft. What I mean is that is that any planes that are/would be developed would be for the airforce because the navy already has their plates full with developing submarines and Surface warships. With that said, my vote would be for navy, because fleet just sounds cooler than squadron
I see what you are trying to get at but the Arbiter is not a singular character, there have been multiple Arbiter's; 192 or something like that. On the other hand Thel' Vadam is a unique character who took a active role in war even before he became the Arbiter.