tolkiens legendarium - What language was the inscription on the Doors of Durin written in?

In the chapter A Journey in the Dark in Lord of the Rings: Fellowship of the Ring, the Company comes to the western gate of Moria, the Doors of Durin, which is sealed and requires a password to be opened. The door carries an inscription that in the book is depicted as follows:

As the picture says, the text inscribed is:

Ennyn Durin, Aran Moria: pedo mellon a minno
Im, Narvi, hain echant; Celebrimbor o Eregion teithant i thiw hin

The Doors of Durin, Lord of Moria. Speak (= say) friend and enter
I, Narvi, made them. Celebrimbor of Hollin drew these signs.

Now that is clearly Sindarin. In fact, it seems to be very bog-standard Sindarin, with no unexpected features. Yet Frodo cannot read it.

That is perhaps not entirely unexpected, since it does clearly say in the caption to the illustration that the inscription is written “in the Feänorian characters according to the mode of Beleriand”, so it could well be like seeing English written using the Russian alphabet to Frodo. (There are indeed clear differences between how this inscription uses the Tengwar and how they are more commonly used in the Third Age.)

But: when Frodo says he cannot read the letters, Gandalf replies:

The words are in the elven-tongue of the West of Middle-earth in the Elder Days. [my emphasis]

In other words, the words aren’t just written in a different alphabet, they’re in a different language. But what language?

As far as I know (and I may be wrong here), all the Ñoldor had adopted (dialects of) Sindarin long before the Second Age when this inscription was made, and the illustration (supposedly taken straight from the Red Book) is in Sindarin also. Frodo knew a fair bit of Sindarin, and considering the entire Company had just come from Rivendell where it is the native language (more so than Westron), it would be very odd if Gandalf referred to Sindarin in such an odd, roundabout way, rather than just calling it Sindarin.

The accepted answer to this related question suggests that the anachronistic use of the name Moria in this inscription may be due to the scribe/illustrator of the Red Book having substituted the older Elvish name Hadhodrond for the more commonly used and widely known Moria; perhaps, then, the entire inscription has been translated into Standard Sindarin from whatever language (presumably closely related to Standard Sindarin, at the very least a common descendant from Old Sindarin) it was originally written in.

Has canon ever addressed this? Did Celebrimbor not speak Sindarin natively?¹ What is this “elven-tongue of the West of Middle-earth in the Elder Days”, then, if not Sindarin? Alternatively, was Gandalf just being inaccurate when he used the term “elven-tongue”? Should he have said “written in the script/characters/style of Beleriand in the Elder Days” or something like that instead?

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¹ I’m aware we don’t know much about his early years, nor even if he was born in Valinor or Middle-earth. The name Celebrimbor is Sindarin, but may not have been the form he actually used himself, similar to how Virgil did not call himself Virgil but Vergilius.

Answer

Well, certainly the words supplied in the inscription, being in Sindarin, are indeed "in the elven-tongue of the West of Middle-earth in the Elder Days":

'I marvel at you, son of Eärwen,' said Thingol, 'that you would come to the board of your kinsman thus red-handed from the slaying of your mother's kin, and yet say naught in defence, nor yet seek any pardon!'

... 'Go now!' he said. 'For my heart is hot within me. Later you may return, if you will; for I will not shut my doors for ever against you, my kindred, that were ensnared in an evil that you did not aid. With Fingolfin and his people also I will keep friendship, for they have bitterly atoned for such ill as they did. And in our hatred of the Power that wrought all this woe our griefs shall be lost. But hear my words! Never again in my ears shall be heard the tongue of those who slew my kin in Alqualondë! Nor in all my realm shall it be openly spoken, while my power endures. All the Sindar shall hear my command that they shall neither speak with the tongue of the Noldor nor answer to it. And all such as use it shall be held slayers of kin and betrayers of kin unrepentant.'

... And it came to pass even as Thingol had spoken; for the Sindar heard his word, and thereafter throughout Beleriand they refused the tongue of the Noldor, and shunned those that spoke it aloud; but the Exiles took the Sindarin tongue in all their daily uses, and the High Speech of the West was spoken only by the lords of the Noldor among themselves. Yet that speech lived ever as a language of lore, wherever any of that people dwelt.

(The Silmarillion, Chapter 15, "Of The Noldor In Beleriand")

From shortly after the time that the Noldor returned to Middle-earth, Sindarin was the only language spoken daily and in public by any of the Elves. So Gandalf is certainly technically correct when he makes this statement.

But Frodo seems at least to recognize spoken Sindarin. When he encounters Faramir and the Rangers of Ithilien, his guards begin speaking to each other in what is almost certainly Sindarin:

They spoke together in soft voices, at first using the Common Speech, but after the manner of older days, and then changing to another language of their own. To his amazement, as he listened Frodo became aware that it was the Elven-tongue that they spoke, or one but little different; and he looked at them with wonder, for he knew then that they must be Dúnedain of the South, men of the line of the Lords of Westernesse.

(The Two Towers, Book IV, chapter 4, "Of Herbs and Stewed Rabbit")

Why then would Gandalf phrase his response to Frodo the way he does?

It seems most likely that Frodo is unable to read the writing because it is written, not in a language or even an alphabet that he doesn't know, but in a mode (that is, a different way of using the letters) he's unfamiliar with: Appendix E of The Lord of the Rings states that

The West-gate inscription illustrates a mode of 'full writing' with the vowels represented by separate letters.

Gandalf, who's undoubtedly familiar with the extent of Frodo's knowledge of Elf-languages, may simply be reassuring Frodo that the words are in a language Frodo does know. There's no way of telling, given that he himself dismisses the wording so quickly.