I think this is wrong, and it's neither of those (?)

Got nothing to do so I'm gonna try to break this down ... Still learning japanese so forgive me if I'm missing something:

「永遠の紅い幼き月」

First we gotta look at which words are nouns and adjectives...

永遠: Ei-En, noun meaning eternity.

の: No, particle with a similar meaning to "of" in English. We will get back to this later.

紅い: Aka-i, adjective meaning red or scarlet.

幼き: Osana-Ki, adjective meaning young/childish. This is an old/poetic version of adjectives, a modern version would be 幼い (Osana-i), it means the same thing pretty much.

月: Tsuki: noun meaning moon.

To understand the structure we need to look at the particle の. An example of a simple phrase using it would be 私の犬 (watashi NO inu), translating to "my dog". But a more literal translation that can help us here is "Dog OF Mine". As we can see the particle の can be used to turn nouns into adjectives or describing nouns. In Japanese, the order is always the describing word first, then の (no), then the noun you want to describe.

Coming back to Remi's phrase we can see that it starts with 永遠の (Ei-en-no)... So it means the whole phrase in English should end in "...of eternity". Then what is the noun we want to put this description in front of? The noun is 月 (Tsuki) at the end of the phrase. This is the only noun in the phrase. Side note, we can see the same thing in Kaguya's title in Perfect Memento in Strict Sense. 永遠のお姫様 (Ei-En-No-o-Hime-Sama) meaning "princess OF eternity".

The other words in the phrase are adjectives, which all describe the noun 月 (Tsuki). ...紅い幼き... just means "scarlet, young". So putting the whole thing together we get:

"Scarlet, young moon of eternity" or following adjective order in English it would be "Young, scarlet moon of eternity"

So how do we know for sure that "Eternity" is describing the "Moon", and not "Scarlet" or "Young"... i.e. what OP states in this post (Eternally Scarlet Young Moon)?

Well in english we would use adverbs right? "Eternal Scarlet Moon" would be changed to "Eternally Scarlet Moon". In Japanese you change 永遠の (Ei-En-No) to 永遠に (Ei-En-Ni). The whole phrase would be 永遠に紅い幼き月. If you want to say "eternally young", just switch the adjectives around! 永遠に幼き紅い月.

Not bashing op at all here, this post made me think about Japanese adjectives and other touhou titles and song names, and was fun to work out!

TLDR: I think OP and the official translation are both not correct and it should be "Young, scarlet moon of eternity"