Wow, what a great double episode about a fantastic story! (And I'm very happy about the hints that your planning to do more of the King in Yellow stories.) I have a lot of thoughts, so I'll try to gather some of them all here.
First of all, Hildred Castaigne: what an interesting protagonist. My first reaction is that he fits well into the types of characters we already discussed with relation to The Frolic and The Insanity of Jones. Has he really had some kind of world-shattering revelation, or is he actually insane? Once again, we have the ambiguity that allows for us to read the story either way. On the surface, I think we're supposed to read it that Hildred is insane: he's already had some problems following his accident and now is relapsing under the influence of The King in Yellow and Mr Wilde. The main clue here is the way Louis dismisses Hildred's tiara and safe. The reader is more inclined to take Louis's dismissal at face value than to accept that Hildred has some kind of secret crown gifted to him by a mysterious entity, and it's a shame we don't get an external opinion on the robes he's wearing at the end to bolster this. If, on the other hand, we want to argue that Hildred has had some kind of revelation, we could point to the fact that the King in Yellow is also described as a king in tatters/tattered king, in which case a trinket kept in a biscuit box would be an appropriate symbol of authority for his chosen representative. The other clue here is in Hawberk's reaction to Hildred's suggestion that he is the exiled Marquis of Avonshire, which absolutely might hint that he knows more than he's letting on. But ultimately I think the way the story is written suggests that we are supposed to read Hildred as insane and not take his perspective at face value (though how much of the story that extends to is an open question).
Based on him being the titular 'repairer of reputations', I actually think Mr Wilde is the central character and crucial to understanding what's going on. We get so little about him that it's hard to say anything for certain (even if this adds to the ambiguous nature of the story): we don't even get a proper explanation of what a 'repairer of reputations' is! Based on Hawberk's reaction to the sign that Wilde puts up, the 'real-world' explanation is that it's just something nonsensical made up by a madman (if Hawberk is just a normal person and not actually the Marquis of Avonshire). The implication is that, as you pointed out in the recap episode, it's some kind of anti-blackmailer, which still doesn't make a whole lot of sense.
What I think is actually going on here is that Hildred and Mr Wilde are not only involved in some kind of revelation/delusion that they share with each other, but that Wilde has drawn a whole bunch of other people into it too (presumably by facilitating their reading of the play The King in Yellow). His clients (the people whose reputations he is supposed to be repairing) aren't who he claims they are. Like Hildred, Wilde has given these people new, important identities in order to give the revelation/delusion a grander scope beyond the mundanity of the lives they all lead. Wilde has convinced Hildred that he is vital to Hildred's mission to take what is rightfully his; with the others, Wilde has convinced them he is the only one who can repair their reputations and made them terrified of the consequences (not just financial) of crossing him. They are all completely cowed to Wilde as the dominant personality within the shared revelation/delusion, except Hildred, who retains some autonomy/authority as the Wilde's chosen heir of the imperial title.
So, it is a shared revelation or a shared delusion? It depends what we make of the other central element: the play The King in Yellow. Is it just a play that drives people insane, or is it the tool of some sort of malign entity that reveals something of the true nature of the universe to those who read it? This and the nature of the documents concerning the Imperial Dynasty of America are part of the ambiguity of the story. It's possible that Wilde is just using these to manipulate people because he has some kind of extreme god complex. But then again, Louis reads the Imperial documents and, even though he dismisses them, there is an implication he sees something there that disturbs him beyond just the ramblings of a madman. Glenn made a good case for there being something fundamentally wrong with Mr Wilde, physically as well as mentally. The cat could be another clue here. Lovecraft uses cats to suggest paranormal/supernatural goings on in his writings, and Chambers could be doing something similar here: is the cat's hatred of Wilde not just a result of his teasing it, but that it knows who/what he really is? It does kill him at the crux of the conspiracy coming to fruition after all. But then why was he keeping the cat in the first place? Nevertheless, we have the possibility that Mr Wilde is some kind of prophet of the King in Yellow (the entity rather than the play).
I already said that I think we are supposed to read Hildred as insane, and I feel like that's the case for Wilde as well: he's insane himself but also manipulating other insane people into sharing some kind of delusion. There are definitely enough clues and ambiguities to suggest the opposite reading though, and I really like the idea that the King in Yellow uses lowly people and things (the 'insane' Mr Wilde, worthless trinkets) to achieve his goals. What those goals are is kind of the big hole in the story if we want argue for the revelation reading though. Beyond just driving people insane, what is the King really trying to achieve? Was this actually an attempt to overthrow the American government and establish the Imperial Dynasty on its throne (eventually leading to the conquest of the world)? If so, it seems a bit feeble to have been stopped so easily. Or is the King just an inter-dimensional anarchist? If so, why target Hildred?
Right, last thing (for now): the world building. I actually really liked this element of the story. I mean, I didn't like the world that Chambers built as such, but I appreciate the effort of setting the scene. (Maybe like Glenn I just have too much of a fondness for world-building RPG books.) I admit that when I read the story I took this straightforwardly as a reflection of what Chambers himself thought would make America a better country (with all that that's a damning indictment of his own political beliefs, from a modern perspective). But I think you made some really convincing arguments that he might actually be parodying that kind of nationalism. I'm not sure about the joke names - I thought they were just an Anglophone author trying to make up some generic sounding foreign names to be honest: the brandy link is intriguing though (and Hawberk is just way too on the nose). Going beyond just this story, it's clear from the others in the series that Chambers had a real fondness for French artistic culture (although there's probably an element of parody there too), so I'm sure he's not trying to claim that getting rid of all foreign influence in America would be a good thing. I also haven't read anything else by Chambers, though, so I can't say anything conclusive.
Actual last thing. On the subject of names, I just want to touch on Avonshire, since Glenn was so dismissive of it. It's absolutely not a nonsense name, and I don't think Chambers was using it as such. The Avon is a real river in England (yes, it's one of our many geographical features that has a name that just means the same thing in two languages), so there are three possibilities for Avonshire. 1) Hildred has a very poor knowledge of English geography. 2) England has had a reorganisation of the counties in this fictional version of 1920 (not impossible, reorganisations happened in 1888, 1972 and 1997), and Avonshire is now a county (presumably) in the West Midlands. 3) Avonshire is part of a secret hidden world, like the Imperial Dynasty of America, and the Marquis of Avonshire is somehow involved in whatever's going on (whether the current holder of the title wants to be or not). The fact that neither Louis nor Hildred can marry the daughter of the Marquis is also intriguing: the ostensible goal is to conquer the world, so why not marry a foreigner? There must be even more layers to this than what the story tells us.
As usual, more ambiguities and questions than answers, but that's the way we like it!