TL;DR: There is no way to tell.
It is a feature of unreliable narrators, that once the reader admits the possibility of their unreliability, there is no firm ground to stand on. If the narrator could be mistaken about one thing, he could be mistaken about anything. The only truth of the matter, after all, is that everything in ‘The Restorer of Reputations’ was invented by Robert W. Chambers, including the narrator. But reading a story as if it were nothing but a collection of lies is deeply unsatisfactory, so it is natural to grope about for some kind of foundation of fact.
When reading a story like ‘The Restorer of Reputations’ we have a choice in how to approach the unreliability of the narrator, once it is suspected or established. For example, we might take the narrator completely at his word and make it a story about the rightful king of future America, foiled in his quest for the throne, and unfairly diagnosed as criminally insane. We instinctively shrink from this interpretation, I think, because in this case the subject matter would be vile. But there is nothing in the text that rules out this possibility.
(An author might, after all, deliberately write such a story in order to highlight the vileness of its literary tropes, as for example Norman Spinrad does in The Iron Dream. But I’m inclined to think that this is not the case here.)
We might, alternatively, decide to interpret direct narration of events as being reliable, while allowing the narrator’s opinions or beliefs to be mistaken. In this interpretation it’s a story set in a future quasi-fascist America whose narrator goes insane due to a fall from his horse and subsequently joins a massive conspiracy run by the blackmailer Mr Wilde. But various aspects of the narrative make this unsatisfactory. For example, the narrator is convinced that his diadem is real:
I lift, from its velvet crown, a diadem of purest gold, blazing with diamonds.
whereas his cousin Louis claims otherwise:
“Come, come, old fellow,” he cried, “take off that brass crown and toddle into the study. Are you going to a masquerade? What's all this theatrical tinsel anyway?”
(In the “take the narrator completely at his word” interpretation this is not a problem because in that version Louis is lying about the diadem in order to convince the narrator that he is insane.)
So the “direct narration is reliable” interpretation is untenable. But if we can’t trust the direct narration of events, then anything in the story could be false. As it says in Wikipedia:
Once Hildred's delusional nature is revealed, it calls into question the other facts of the story, which may all be delusional, as well. For example, the Prussian-style military parades could merely be police on patrol, and the suicide booths may simply be subway entrances.
In this interpretation, the ‘clues’ identified in the post are just further parts of the delusion: perhaps Hawberk lies about finding the fragments of armour in order to appease the dangerous narrator; and maybe the narrator is mistaken in his interpretation of Hawberk’s and Constance’s reaction to hearing the title “Marquis of Avonshire”.
Critic Wayne Booth wrote about the problem of unreliable narrators in The Rhetoric of Fiction, with regard to the works of Henry James:
No amount of care, no amount of intelligence, no amount of background reading, can yield the kind of security about ‘The Liar’ that all readers can feel about ‘The Beast in the Jungle.’ [… A] story can hold together only if such perplexities are kept within certain boundaries—wide as those boundaries may be. Our very recognition of complexity depends upon the clarity of our vision of the elements which go to make it up. […] Once on this road we cannot turn back; we cannot pretend that things are as simple as they once seemed.
‘The Restorer of Reputations’ has perplexities that go beyond all reasonable bounds. But a story in which everything is delusional is just words on paper and not worth the reading. So I think most readers will be driven to take a middle view in which much of the narration is reliable, but not all, and they can’t be sure exactly which parts. This uncertainty and unease about what’s ‘real’ and what’s not contributes to the uncanny atmosphere of the story.