When reading that bit one of my first ideas was making a "Board of Governors" of the entire city (aint no rule the council has to consist only of those families, it has to be unanimous or the board have any other power). Carliss's situation stops that from working unless there's some distant relative who has a claim in the city, but I doubt Cheliax would go for broke on the chance one line is truly extinct.
There isn't a rule that anything has to be done a certain way. You can change things any way you want as it's your game.But, if you're going exactly by what the books say, page 23 of book 5 tells you that that the responsibility can be passed on to a legal heir even if the person isn't of the original family's blood (otherwise Laria wouldn't be on the Board), it also says that the vote doesn't have to be unanimous, and that House Thrune consults their copy of the Cheliax Covenant (which requires getting another copy of it from Odexidie as Queen Abrogail I destroyed House Thrune's original copy), and consult mortal and infernal barristers to confirm that what the PCs/Lord-Mayor are saying is correct.
Having the entire city vote could be problematic for the contract as it would mean the PCs likely don't have to go visit Odexidie, the language involved would have to be incredibly specific as to who counts as "being part of the city," and that set-up likely wouldn't have worked for the original intended purpose as Abrogail I intended to convince or force the Board of Governors to ratify her as the Lord-Mayor and doing that tho the entire city would be harder (if not impossible) and would take too long if she was in a rush to exploit the loophole.