Far be it for me to defend the authors of T4 (though I am a big fan of Greg Porter), I disagree with you, Mike, about the appropriateness of the Arlea Committee to the setting. I think it functions as the politburo of the old USSR or in the current PRC where you still have a totalitarian dictator but there remains other powerful interests that 1) must be placated, and 2) from whom future leadership can emerge.
Military dictatorships may not need a politburo (at least at first), but the Third Imperium was not a result of a military coup, it was a diplomatic/political coup that subverted the former Sylean Senate. How do you get the most powerful economic/political families of the prior regime to willingly (bloodlessly) choose an Emperor? You give them a piece of the governing action to protect their interests.
As far as Houses, it seems we are quickly getting into conjecture but we do know for sure that 2 of your 4 names are out and in my view no better than 1 for 4, Mike. Neither a House of Makhidkarun nor Naairka (which would be House Igsiirdi) were part of the Imperium at its founding. They came later when Vland joined, so they can't be part of the original nine signatories to the Charter of Sylea. We are told that explicitly in TD18 and it is implicit in M0.
That same fact may argue against House Shiishuginsa, the founders of Zirunkariish, but I'm with you and don't think it does. Zirunkariish is a megacorp in M0 and it is the ONLY financial megacorp on the list. It is hard to have a growing, high-tech, trade-based society without finance. Zirunkariish finances Vland, but does so AFTER contact with Sylea, which to me is a clue that they come from Sylea. So I am with you and number House Shiishuginsa with the original 9.
As for House Alkhalikoi, I'd say there is explicit evidence in CT they are not part of the original Arela Committee. Arbellatra is the last Emperor of the Flag and according to CT, it is the marriage of her son Zhakirov that "cemented an alliance between the Alkhalikoi dynasty and the business interests of the Imperial Core…" The whole early history of dynasty is of an outsider that takes over, not an insider rising to power.
I don't know what your argument against House Ling would be, Mike. Ling Standard Products is a M0 megacorp. Obviously they have a Solomani name, but the Ling must have emigrated long ago to have achieved such a status. LSP is explicitly indicated as one of 4 megacorps in Sylea along with GSbAG, Turkera, and Zirunkariish. (In any event, they certainly can't have been based in the Solomani Rim as that area is not reincorporated into the Third Imperium until beginning in the 5th century of the Imperium.)
So I think we are up to 5 because we can add House Shiishuginsa of Zirunkariish.
Need 4 more.