My feeling on all this would be that the Lord of the Rings is shot through with imagery and and patterns of thought that echo traditional Catholicism much more strongly than other varieties of Christianity. Which is what you would expect of a Catholic, of Tolkien's generation especially.

So for example the hymn to Elebereth is very reminiscent of Catholic hymns to Our Lady

Snow-white! Snow-white! O Lady clear!
O Queen beyond the Western Seas!
O Light to us that wander here
Amid the world of woven trees!

Gilthoniel! O Elbereth!
Clear are thy eyes and bright thy breath,
Snow-white! Snow-white! We sing to thee
In a far land beyond the sea.
O stars that in the Sunless Year
With shining hand by her were sown,
In windy fields now bright and clear
We see your silver blossom blown!

O Elbereth! Gilthoniel!
We still remember, we who dwell
In this far land beneath the trees,
Thy Starlight on the Western Seas.

Compare tyhis to the mediaeval Salve Regina:

Hail, holy Queen, Mother of Mercy!
Hail, our life, our sweetness, and our hope!
To thee do we cry, poor banished children of Eve,
to thee do we send up our sighs,
mourning and weeping in this valley, of tears.

Turn, then, most gracious advocate,
thine eyes of mercy toward us;
and after this our exile
show unto us the blessed fruit of thy womb Jesus;
O clement, O loving, O sweet virgin Mary.
Pray for us, O holy Mother of God
That we may be made worthy of the promises of Christ.

Or again compare it to what I think is the loveliest of the Mary hymns, written in the 18th century:

Hail, Queen of Heaven, the ocean star,
Guide of the wand'rer here below:
Thrown on life's surge, we claim thy care -
Save us from peril and from woe.
Mother of Christ, star of the sea,
Pray for the wanderer, pray for me.

Sojourners in this vale of tears,
To thee, blest advocate, we cry;
Pity our sorrows, calm our fears,
And soothe with hope our misery.
Refuge in grief, star of the sea,
Pray for the mourner, pray for me.

To save myself typing out the Elbereth hymn I used a search engine to find someone who had already done that, and found this essay about this very thing, which makes interesting reading (though I'd distance myself from some of the writer's views).

And in relation to Tolkien and "fundamentalists", I would expect that these very aspects of his writing would in themselves be sufficient reason for some varieties of "fundamentalists" to see the book as unwholesome and dangerous.