invented to keep out the cold
It seems that Cheliax isn't trying to do any sort of political dealing here, not if they're actively claiming the throne. This is obviously modeled off Queen Domina's technical usurpation of the throne, it's not actually as unprecedented as the paper makes it seem. This is why you don't trust people just because they write things down. Queen Domina made her deals from a place of power, though. She funded the Asmodean expansion only after successfully increasing Korvosa's trade. If this Ignasi got the Most High to cast him a Gate to bring him here, he is much more indebted than she was. Debts owed to the Asmodean church are risky, and debts owed to whatever forces in Cheliax can get Gates on command are likely riskier still.
Still, the memory of Queen Domina is strong. She did rule well, she was not too long ago, and many people remember her fondly. They'll be trying to play on that, remind everyone of what happened to the last family that tried to oppose an Arabasti reclaiming the Crimson Throne... He definitely has a chance here. And how strongly do the Houses actually want to oppose him? Ex-House Porphyria is a rather memorable warning.
For now, she can't do much with this, though. Vague ominous warnings would be a bad idea with no actual grounding in knowledge, and she doesn't know where they would be effective. Maybe if she sees something she can amplify it. Until then... more guard shifts.
(Unless she has mail, but that seems unlikely unless the Arkonas really want to know more about the Mantis. And even then, Captain Paras would be a better source.)