My guess is that you should take some governmental similarities from the United Citizen's Federation of Starship Troopers.
That would be a great idea, initially the majority get but over time it is only those who served that can rule very Clans but with a Starship troopers twist
As for the name Star League in Exile does have one issue for me it's got the name Star League in it.
I know they are the Star League Defence Force but calling themselves the Star League screams having a First Lord which could bring tensions which is why I think Kerensky should make an announcement: The Kerensky Proclimation
"My fellow citizens of the Star League we have come a long way. We have turned our backs on the petty differences of the Succession Lords to focus our Defence Force on its primary objective: Peace.
The Defence Force was created to protect humanity from wars that would bring it to the edge of destruction, wars that only madmen could envisage. We, together, defeated the Usurper and we together travelled to these Pentagon Worlds which we call our own.
With this Proclimation I and the members of the High Command have decided that there will be no First Lord for our Defence Force until we return to our rightful place defending Terra.
However ours is an Army and Navy without an enemy. We can tame these Pentagon Worlds but what next you may ask? Why serve? With this decree I write into law that everyone that can hear my voice today is eligible to be a Star League citizen and going forward all members of the Star League Defence Force will be considered Citizens with all the rights attached.
I have also ordered that our Army and Navy create Exploratory Divisions. These will explore deeper into the Periphery and discover more worlds for us to exploit and colonies. We will also continue to do what our Defence Force was created to do and defend our people by patrolling the Exodus Road and nearby Periphery realms. All threats to our population will be dealt with severely as we are the Star League Defence Force and this is OUR Star League.'