The biggest problem here is that English has no synthetic case.
The preposition "of" is analytical without a case ending.
The novel gate-suffix derived analytically from the Watergate scandal is fusional, cp. gamergate etc.
Enclitic 's is definitely fusional. Thus, yours displaced youren. The formation by analogy is incomplete, so for example own remains in place (Old English Godes āgen bearn “God's own child”), genitive -es was removed (already in OE weakly inflected āgenan beside āgenes), accusative was merged into dative (OE hine "him") and it is widely recognized that reconstruction of pronouns and sentence structure is difficult (hence any mistakes in the above are my own).
Beyond this, Personal names may be subject to inflection. Henceforth, any inflection was either fossilized in the name (e.g. French Disney, d'Isigne "from Isigne”) or removed (e.g. Frakfurt[er]). This can become very confusing when the morphology of the name is less than clear e.g. Walt, Walter, Dalt, Dalton etc. (cf. -ton, de.Wikipedia). An ambiguous case remains in the Daltons, I stayed Hilton's', at the Daltons' (one might argue that the apostrophe is not pronounced either way, "James's Park", "King Charles's Island" but more often "King Charles' activities", Jameson showing it's Norse or Norman heritage, Charles being clearly Frankish in origin, Latin Carolus).
The Saxon Genitive or „apostrophic genitive“ is always definite, “Peter's car”, whereas indefinite constructions may use “of”, which can be used to show that the Saxon Genitive in English is not a functional case ending; e.g.:
[[the king of England]’s horse]
[one of [Peter's cars]]
slightly ambiguous as a placename e.g. King's Cross (London, north of Charing Cross), Kings Cross (Sidney).
(de.Wikipedia: Sächsischer Genitive
It is also notable that the inverse, # King [of] England['s], is abnormal in English except for a few higher register constructions, e.g. the Era McCarthy, often as Anglo-Norman heritage, Attorney General (but District Attorney).
Hence the question does typically never arise, if “the Screwtape Letters” ought to be genitive. Neither is there a denominal adjective, not Japanese but the Japan files.
In grammar, a noun adjunct, attributive noun, qualifying noun, noun (pre)modifier, or apposite noun is an optional noun that modifies another noun; functioning similarly to an adjective, it is, more specifically, a noun functioning as a pre-modifier in a noun phrase.
(https://en.wikipedia.org/wiki/Noun_adjunct)
This argument is often made in lexicography to avoid redundantly defined adjectives in the lexicon, since every noun can be attributive.
The bulk of my answer is concerned with showing that -n would be historically correct, as it is in German following prime evidence of Sonnenschein "sunshine" (“from heteroclitic inanimate Proto-Indo-European *sh₂wen-, oblique of Proto-Indo-European *sóh₂wl̥ (“sun”).” en.Wiktionary: sun).
We see that this impression is misleading and mainly ruled by euphonic principles as Japan < 日本 (Jp. Nihon, Cantonese jat6 bun2; Malay Jepun etc.) has nothing to do with case, at all. Eventually, it seems that the Saxon Genitive is no more a genetive case. It only looks like one. In this view, the question for accusative, dative, or even ablative interpretations can be ruled out.
The remaining question should be, why letters presents as plural when briefing or causa do not, but the answer is trivial enough, in which letter is reinterpreted as count noun (it should be prudent, however, to attempt unconventional comparisons as long as a large part of the etymology remains gappy).
Anyway, Latin genitive ablativus postposition construction may show that the genetive does not mean what you think it means: [urbis] causā “for the sake of [the city]”, indeed for Peet's sake, that is a pleeding in the case made by Peet (as often as his name is invoked he must have made a darn good case too).