So according to a news article from the period in Dawn of the Jihad, the Second Star League abruptly dissolved thus:
As the conference started, the Capellan Confederation sent a note announcing their withdrawal, while the Lyrans and Suns declared their withdrawal in person.
The other house lords objected to the Capellan, Lyran, and Suns decision to leave and called for a vote of no confidence in the Star League. The Taurians and Word of Blake were admitted to vote in this, and the result was 6-3 in favor of retaining the League, which failed to achieve the necessary 3/4ths majority to keep the League going.
(Sarna says the vote of no confidence was actually called by First Lord Mansdottir, who hoped the council lords would vote to retain the League, but I can't find any reference for this. Dawn of the Jihad suggests that all the other house lords demanded the no confidence vote)
From these data points we can assume that the three houses that had just announced their withdrawal were permitted a say in the no confidence vote (presumably their membership actually ended with the end of the council), and all three of those states voted to dissolve the Star League, but the remaining states uniformly voted for the Star League to remain.
Which raises my question. If they wanted the Star League to remain WHY DID THE OTHER HOUSE LORDS CALL FOR A VOTE OF NO CONFIDENCE IN THE FIRST PLACE?
It's simple mathematics. They apparently needed a three fourths majority to keep the league going. There were nine votes. They knew the three states that were on the way out were going to vote against it. They therefore knew they could only ever get six votes in favor of sustaining it. The only way the vote of no confidence could fail and the league could be sustained would be if one of the states that had just decided to leave the Star League suddenly went "Oh gee, I know I just said we're leaving the Star League, but you guys should totally keep it going."
So why did everybody who wanted to keep the League decide to put it up on the chopping block in the first place? What am I missing?
Much of it is "Power of Plot"-but not all.
You've got to understand, an alliance of victors tends to fall apart anyway, and everyone who said they wanted a Star League, left out one important aspect: They wanted the original, with themselves in charge.
2nd Star League was constructed in a way that that literally couldn't happen.
That's the first part, the second, was how easily Sun Tzu corrupted it during his tenure, and how ineffectual the alliance was in preventing the very thing it was supposed to prevent. A historical example you might consider, is the League of Nations, which utterly failed to keep the peace-a peace it was explicitly created to keep.
2nd Star League, after Bulldog and Serpent, was incapable of doing...well...anything.
The unity of purpose wasn't there and the external threat was....well, gone.
The boogeyman of the Clan invasion was mitigated in the minds of not only the public, but the governments involved as well. It didn't matter than 40% of the Lyran state was still occupied, because the FWL, CapCon, Etc. weren't under threat and they knew it, and if there's anything to be doped out of the FedCom civil war, it's that the Lyrans and Feddies would rather throw their key units up against their own key units in a family spat, rather than fight off a foreign invader with a dogma of social conversion.
(this has historical precedent too; Chinese fought Chinese much harder in WWII than either major faction of Chinese fought the Japanese invaders.)