The Swindlers ("bedrager, in Danish) who fool the Imperial Court into believing that their supposedly invisible clothes are actual delicate garbs and not purely fictitious are described as sagisha (詐僞者, "prevaricator") in Watanabe and katari (欺騙者) in Takahashi's version. The latter using the non-standard reading katari (swindler) with the kanji usually read as kihensha ("deceiver") thus to highlight their duplicitous nature.
Their speech patterns are notably courteous in the latter, but utterly levelled out in Watanabe's as noted due to his "direct translation" style into a more neutral Classical Japanese, as such I shall be omitting that part of the analysis here and instead focusing on how Takahashi treats these characters.
One key-scene that may illustrate how decorous the Swindlers are when they humbly ask the Emperor to undress so that His Majesty may don the invisible clothes.
"Denka moshi gyoi wo dasshitamawaba, shinra oosugatami no mae nite kono ishou wo otsuke mairamashou"
"Mayhaps, if Your Highness graciously deigns to undress his honoured clothes, that we, lowly servants, may in front of the great dresser (= mirror) garb you most humbly in your august robes."
The archaic keigo is present, tamau (the nobleman grants) in its conjectural form tamawaba (if the nobleman were to grant [us the favour of]), here coupled with dasshu (to undress) to express the most courteous request. Tamau is quite essentially Classical Japanese.
They refer to themselves with shinra (we lowly servants/menials), an extremely archaic humble way of self-reference hearkening back to the ancient Chinese Imperial courts where 臣 (Middle Chinese was used as a humble first person pronoun by the courtiers and servants of the court, literally meaning "minister" or "courtier," but here highlighting their lowly office compared to that of their ruler.
They frame the petition to clothe the monarch in the non-existent clothes with otsukemairu, lit. "to conduct the honourable clothing", where mairu is the humble version of verbs denoting motion (iku, kuru) or action (suru, yaru), here in the text itself render in the volitional form ~shou, and by prefixing o- to the conjugated verb they render it an honorific one and express the utmost humility, well knowing that they are in fact deceiving the Monarch.
Note that they refer, and as does the rest of the cast, to the King (Takahashi's rendering of the Emperor) with denka, Your Highness, the style of a prince or lesser ruler than that of heika (陛下, "Your Majesty"), the proper style for a king or emperor. This may not be case of belittling the Andersen Emperor, but merely following contemporary protocol or conventions at the time, I am unfortunately not all too versed in this matter at this writing hour.
They also use wareware-domo (我々ども) in the very beginning of the story, which in an incongruous, knowingly so, way combines the formal wareware (we) with the humble plural suffix domo, thus appearing neutrally formal and humble at the same time, not really being arsed to use watakushi or temae, which are the proper humble pronouns to use the plural suffix domo.
When these Swindlers are ennobled by His Majesty they are in the original called Vævejunker ("Sartorial Junker") in Watanabe's version given the grand title of Teikoku Choutei Orimonoshi (帝國朝廷織物師, "The Emperor’s Weaving Master for the Imperial Court") and Takahashi merely states: kunshou wo tamawatte kizoku (勲賞を賜はって貴族, "awarded them the title of nobility"), thus skipping the whole process of translating their new title of office - though he and another translator, as we shall later see, are the only ones to omit the name of the office.
Next, we shall look at key scenes.