Tolkien was certainly a “traditionalist”, if by that term we mean a defender of things primordial, and therefore timeless, and hence a critic of the modern world—its philosophy, art, and politics. This is implicit in The Silmarillion and The Lord of the Rings, of course, but it’s also explicit in his hugely important article “On Fairy-Stories” (see Essays Presented to Charles Williams, edited by C. S. Lewis), where he argues that fantasy, of all forms of art, serves best to satisfy certain of man’s “primordial desires”, among them “to survey the depths of space and time” and “to hold communion with other living things” (44). Consider, as just one example, the following autobiographical passage from this excellent essay:
“Not long ago I heard a clerk of Oxenford declare that he ‘welcomed’ the proximity of mass-production robot factories and the roar of self-obstructive mechanical traffic, because it brought the university into ‘contact with real life’. He may have meant that the way men were living and working in the twentieth century was increasing in barbarity at an alarming rate, and that the loud demonstration of this in the streets of Oxford might serve as a warning that it is not possible to preserve for long an oasis of sanity in a desert of unreason by mere fences… I fear he did not. In any case the expression ‘real life’ in this context seems to fall short of academic standards. The notion that motor-cars are more ‘alive’ than, say, centaurs or dragons is curious; that they are more ‘real’ than, say, horses is pathetically absurd. How real, how startlingly alive is a factory chimney compared with an elm-tree: poor obsolete thing, insubstantial dream of an escapist! (77).
It’s perfectly obvious that an author who looks at the world in this way would have sympathized with much in Guénon’s Reign of Quantity and would have read with understanding and delight the chapter “Fall and Forfeiture” in Schuon’s Light on the Ancient Worlds. But whether Tolkien actually did read these or other Traditionalist authorities I frankly doubt.
You are right, of course, that C. S. Lewis was introduced to Guénon by his student Martin Lings, but according to Lings Lewis was very much put off by what he found—especially by Guénon’s apparent disparagement of the ancient Greeks (excepting Plato), whom Lewis dearly loved—and his response to the books was severely negative, though I do admit (and have sometimes argued) that That Hideous Strength reflects an appreciative reading of The Reign of Quantity. Nonetheless I think it highly improbable that Lewis would have mentioned, or certainly recommended, Guénon to Tolkien, and I can’t imagine how otherwise Tolkien would have become aware of the Traditionalist School. I suspect the parallels you rightly note are owing simply to his being steeped in medieval language and literature.
Was Tolkien a mystic? I’m certainly intrigued by the evidence you offer from one of his students, Simonne D’Ardenne, whose question, “You broke through the veil, didn’t you?”, elicited the answer, “Yes, I did”—though I would wish to know more about the context of this exchange so as look more closely at the “veil” of which they were speaking. Be that as it may, I see no reason not to think that the “sub-creator” of Middle-Earth was simply being candid when he claimed that much of his writing was the result of “inspiration” and not “invention”, an inspiration that may well have led to his passing through a “door on Other Time” and thus to his being permitted at least a glimpse of something “outside Time itself” (to quote once more from “On Fairy-Stories” ).
Let us do remember, however, that a mystic is not, or not necessarily, a gnostic or jnanin, and a traditionalist, at least as I’ve used that word here, is not, or not necessarily, a perennialist. I’ve no doubt at all that Tolkien the traditional Catholic, who was more than a little scandalized by Charles Williams’s ideas, would have been even more horrified by the notion that there is more than one saving Path to the Summit.