We know what happened to Kerensky and his descendants in canon but let’s for the sake of this discussion say that the Clans don’t happen. Instead, in this thought experiment I'd like to discuss what a Star League in Exile (or the Exodus Planetary Council if you prefer) would look like and what changes might have to occur in order for it to endure. The following is what I think but I'll leave it up to you to expound on your vision.
An aging Protector Aleksandr Kerensky acts to establish a new Unity Council early in the 2790s as old allegiances began to revive. This successor to the High Council represents each of the old Great Houses which once constituted the great Star League. Carefully selecting candidates who best represent a united Star League in Exile government he successfully co-opts much of the growing opposition. Kerensky and his heirs naturally will serve as representatives of the Terran Hegemony on the Council as well as the Protector. The only other two candidates I have in mind are Raymond Sainze for the Draconis Combine and Kailen Steiner for the Lyran Commonwealth. Sainze is the epitome of the samurai culture while Steiner is the closest thing to a legitimate heir to the Lyran throne. I don't have faithful representatives of the Federated Suns, Capellan Confederation, and Free Worlds League yet but I will list them as they come to mind and I'm open to suggestions. Also, I'd like to add a seventh member of the Council representing Spacers and the Periphery. I figure these Outsiders who live on the fringes of the Star League in Exile would make some sense, but I see resistance here. Distrust of the Periphery probably still remains high and I'd imagine few of their denizens actually made the journey with the SLDF so their numbers would be quite low. However, I feel like there was always a movement to add a Council seat to represent them. Spacers give us that chance as they are a nomadic lot and could perhaps be led by a figure like Janos Grec to establish together with any loyal Periphery denizen a new Councilor.
Secondly, after the establishment of the Council I'd hit on military reformation. It is said that the SLDF was reduced from 2 million and half a million (one-quarter). I still have to do the math on this first great demobilization which of course makes sense but it gave me an idea especially with all the discontent it created. If we establish a new 13 to 12 ratio from that original 25% we can get a get number for 'House militias' to defend and police their own populace. Each 'House' would receive 2% of the League's overall defense budget to fill out their own army and navy as they see fit. Again, the aim of this would be to mollify critics who could be remilitarized in 'House units'.
Third, get an effort going to keep an eye of the InnerSphere in order to report back on the Succession Wars. Be a boon for propaganda if you could show your populace the mess that they've left behind.