The cross-pollination between "fey" and "fae" is complicated, and the actual origin gets fairly murky. However, the sense of "fey" as implying that someone is acting not like their usual self, and frequently obsessed with something or someone, or overcome by an emotion to an unusual degree, is quite old. It's easy to supposes that either possession by, or replacement by, some other-worldly or supernatural entity is alluded to, but it's apparently not linguistically clear how and in what order "fey", "fae", etc. acquired which of their various meanings. Some early examples that seemed interesting:
1856 J. Ballantine Poems 207 "Wad ye rax his craig, When our daughter is fey for a man?"
1921 ‘M. Corelli’ Secret Power iii. 34 "‘But I was “fey” from my birth—.’ ‘What is fey?’ interrupted Miss Herbert… ‘It's just everything that everybody else is not’—Morgana replied—“Fey” people are magic people; they see what no one else sees,—they hear voices that no one else hears—voices that whisper secrets and tell of wonders as yet undiscovered.’"
That said, DF draws strongly from Tolkien in many places, so that is likely to be the proximate source for "fey mood". Note that Tolkien's use of "fey mood" in particular, and "fey" in general, seems to lean toward situations involving battle-rage or death; either obsession with it, or determination to do something despite the chance of death.
"Dread was round him, and enemies before him in the pass, and his master was in a fey mood running heedlessly to meet them.
Tolkien, J.R.R.. The Lord of the Rings: One Volume (p. 724). Houghton Mifflin Harcourt. Kindle Edition.
"Alas that a fey mood should fall on a man so greathearted in this hour of need! Are there not evil things enough abroad without seeking them under the earth?"
Tolkien, J.R.R.. The Lord of the Rings: One Volume (p. 798). Houghton Mifflin Harcourt. Kindle Edition.
"He stood a moment as a man who is pierced in the midst of a cry by an arrow through the heart; and then his face went deathly white, and a cold fury rose in him, so that all speech failed him for a while. A fey mood took him. ‘Éowyn, Éowyn!’ he cried at last. ‘Éowyn, how come you here? What madness or devilry is this? Death, death, death! Death take us all!’ Then without taking counsel or waiting for the approach of the men of the City, he spurred headlong back to the front of the great host, and blew a horn, and cried aloud for the onset. Over the field rang his clear voice calling: ‘Death! Ride, ride to ruin and the world’s ending!’"
Tolkien, J.R.R.. The Lord of the Rings: One Volume (p. 843-4). Houghton Mifflin Harcourt. Kindle Edition.