The most significant obstacle to the Crown Loyalists unquestioned grip on the throne is the opposition movement organized around and spearheaded by House Suroth. Lady Suroth's backers, aside from her own House and associated minor houses, consist at this point of House Danaani and the Free Cities of Valencia and Gotham.
Against the glittering array of the Crown Loyalist's massive coalition, this would seem like a paucity of resources, especially considering Suroth's lack of Primal artifacts; House Suroth's own Logrusfoil is held by their principal enemy in this affair, House Hendrake, and House Danaani's Logrusfoil is held by Princess Asuka of Amber.
However, what they lack in quantity and in powerful artifacts, Suroth makes up for in quality, organization, and leadership. While House Suroth did indeed suffer during the widespread devastation of the Logruswrack, it was devastated to a much smaller degree than any of the other Houses, as their proficiency in the shifting arts leant them much protection. Their great warriors, leaders, and sages for the most part all yet live. While they lack such potent individuals as Loewen or Touga lending them their might (though Lady Suroth is a force not to be reckoned with lightly) House Suroth's military supremacy is currently such that they could fight most combinations of the Great Houses with an excellent chance of winning. Only House Fomorian (and Darios, if they fully reunite) can even hope to match their armies alone, and even those two fall well short. This is a very significant advantage, and in fact would almost seem to be decisive.
And yet. House Suroth lacks the large degree Amberite backing that the Loyalist faction has, and the power a Prince or Princess of Amber brings to a cause, to say nothing of the power of the kingdom as a whole, is undeniable. To be sure, Suroth is not without total lack of recourse here; Princess Kuonji and her son Prince Setsuna are members of House Danaani, and can be presumed to (and in fact have a legal obligation to) respond to a call to fight from them; however, they both have far, far stronger ties to Amber than to Chaos, and Kuonji in particular is known to have extremely strong ties to Prince Touga, who may have strings tied to her he can pull. Thus, Kuonji and Setsuna can hardly be considered the most vigorous and devoted of allies.
Some few other members of the line of Amber have expressed quiet approval of Lady Suroth, and Shigure Souma, the Chaosian Ambassador to Amber, is one of her bannermen and is known to have the ear of King Tylor. Other members of House Souma have had extensive dealings with Amber and its members are well-known and well-liked among its Royal Family. However, aside from these few contacts, there has been little in the way of real bond-building between Amber and House Suroth, so an Amberite thumb on the scale does not seem likely to come down on Suroth's side, which fact, taken in concert with the numerous and powerful Amberites working for the Loyalists, represents a great check of Suroth's power.
Lady Suroth is a legitimate child of a legitimate child of Dios, one of only two legitimate grandchildren of Dios (the other being Jack) who is both a legal adult and has assayed the Logrus. The legal basis of her claim to the throne is that, since Azalyn had not assayed the Logrus at the time of her fathers death, she has no claim, and the succession devolves upon the next person in the line, namely, herself. She has thus petitioned the College of Heralds, on these grounds, to formally recognize her as the Queen of Chaos.
Like Azalyn's claim, this argument is uncertain at best, given the utter lack of legal precedent in this area.
Suroth's opposition movement has promulgated a political platform based primarily upon scrapping the current confederation-style government of the Courts, with a weak King and the Great Houses possessing considerable independence and sovereignty, in favor of a more unitary system of autocratic monarchy, stripping the Houses of much of their autonomy and concentrating the power in the hands of the head of state. She contends that this is both necessary and desirable, to help ensure Chaos' continued survival as an independent power, to eliminate the constant internal squabbling that only serves to make the Courts weaker than they might otherwise be, and to better provide for the common defence and security of Chaos as a whole.