He was, for example, a philologist, yet non-philologists enjoy LotR, as Tolkien hoped they would.  He loved non-Christian mythologies, and wished that more of the non-Christian Anglo-Saxon mythology had survived.  Catholics are not the only people interested in death and immortality, the self-destructive nature of evil, the hierarchical structure of authority, and the morality of serving a rightful leader merely because (s)he is the rightful leader.

Only in his letters does Tolkien claim that LotR is a fundamentally Catholic work.  He seems to seize on any inference of Catholic influences, while he often becomes defensive about any inference of non-Christian influences.  This results in his agreement that Galadriel is based on Mary, Mother of God, despite obvious differences.  It also results in denials that LotR is at all influenced by Norse or Anglo-Saxon mythology, despite what seem like obvious influences.

Tolkien did try to make LotR consistent with his Catholic beliefs, mostly by removing any overt reference to religion.  He had much less success making The Silmarillion consistent with his Catholic beliefs, since much of it was all about religion.  And even in LotR, W.H. Auden made a telling point when he wrote to Tolkien and wondered whether the orcs, a seemingly irredeemable race of sentient creatures, were not fundamentally inconsistent with Catholic beliefs.  Auden probably never knew how hard Tolkien tried to resolve this seeming heresy in his unpublished writings after LotR, with little success.

So despite Tolkien's claim, I disagree with your premise that LotR is a fundamentally Catholic work.  And even if Tolkien's Catholicism heavily influenced the work, even he would admit it is not an overtly Catholic work.  Tolkien designed his story for readers of all faiths.

Furthermore, although I do like to look for the author's intentions in a book, there is nothing wrong with readers projecting their own likes and dislikes onto a novel, painting, play, or movie.  Any book as popular as LotR is bound to be interpreted differently by different people, and liked for many different and often inconsistent reasons. 

For example, Tolkien left out overt references to religion because he was religious, but for that very reason many non-religious people enjoy the book.  Tolkien was so reactionary that he preferred feudalism to capitalism, but for that reason LotR appealed to many leftist hippies also disenchanted with capitalism.  Tolkien so hated 20th-century warfare that he escaped into a truly virtuous war in a fantasy world, but for that reason LotR appeals to many conservative "hawks" who believe that warfare can still be virtuous today.  In short, Tolkien would be appalled at the many and various interpretations of his story, and indeed was appalled during his lifetime -- but any book as popular as LotR is bound to be interpreted differently from how the author intended by most of its readers.

“I dislike Allegory - the conscious and intentional allegory - yet any attempt to explain the purport of myth or fairytale must use allegorical language.  (And, of course, the more 'life' a story has the more readily will it be susceptible of allegorical interpretations: while the better a deliberate allegory is made the more nearly will it be acceptable just as a story.)” (From Tolkien Letter # 131.)

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