Well, first off:
... admittedly, any Interactive AAR would be interesting for me.
That being said, merchant or Hawaii style capitalist republics are definitely intriguing. The Confederate States could also be interesting, especially in a case where they would probably eventually lose (so instead of Presidents or Shadow, where the nation is getting stronger and stronger, the nation is slowly caving away as political bickering and hasty plans are set up).
I've also always wanted to see (and considered starting at times, but never found the time to) a Chinese Warlord Era AAR. The era simply is so much like an interactive AAR - Yuan Shikai allies with the KMT for awhile then attempts to form a monarchy of his own, Cao Kun blatantly bribes his way to the presidency...
The Emperor (to begin with) and the national government (once it is set up) could be NPC. Participants would be focusing on regions and cliques - trying to expand their influence and the influence of their allies (in political blocks such as the Communists, the Kuomintang, the Beiyang, etc.). There could be coups or wars - either player initiated as in SOTA or only started by the events of the story (so if a clique was distancing itself too far from the national government, the national government could declare an expedition to bring them back in line). Instead of primaries, each faction could set up its own "national government" if powerful enough (enough players and perhaps control of a major city to be the capital) and vote in that like in a primary (on NPC's politicians and legislation). Instead of national elections, each clique would decide how much it would support each of these governments (in-game in terms of soldiers, money, and obedience to legislation). Failure to support or to follow legislation could lead to fighting or lack of protection from the other cliques.
It would take some work to figure out a not-overly-complicated model, but it could be very cool in the end - lots of different government types, all sorts of economic and military policies and shenanigans possible, etc.