From an interview I did with the late Dr. Michael Stanton back in 2004:

One thing I have noticed is that even though Tolkien coined the term back in the late 1930s, when he was still in the early stages of writing LOTR, very few of his earlier critics seem to have picked up on it. Paul Kocher (1972) devotes a page or so to the concept; Randal Helms (1974) hardly more. The first extended treatment I have found was in Ruth Noel’s Tolkien’s Mythology (1979); by the time we get to Tom Shippey’s Tolkien: Author of the Century (2000) we find half a dozen or more pages. Maybe it’s the New Age influence which has brought the quasi-spiritual side of Tolkien into prominence. You could probably deal better with that question than I.

“Eu-cata-strophe” as a word: always important to remember that Tolkien was first and foremost a student of languages, so what did he mean by this coinage?

“strophe” = turn
“cata” = down, against, back
“eu” = good

In our modern usage “catastrophe” has come to mean a large-scale calamity or horror (Hurricane Katrina comes to mind); thus Tolkien’s word means almost the opposite: a large-scale turn toward the good from an originally dire situation. What Tolkien says about it in his discussion at the end of “On Fairy-Stories” is that it is a kind of revelation, a glimpse or feeling of joy not unmixed with sorrow (the two always seem intertwined).

What Tolkien seems to suggest is that “drama” and “narrative” are two independent ways of relating human actions: one by representation, the other by story-telling. He says that “tragedy” is the natural form of drama, whereas “eucatastrophe” is the natural outcome of a told story. an achieved and merited happy outcome, earned by moral qualities like courage and loyalty.

I think it’s important to keep in mind that a “eucatastrophe” does not mean a happyending: as Tolkien says in his essay, fairy tales have no ending (“they lived happily ever after” being just a dodge or a time-saver) and as Sam says in LOTR itself the old stories just go on and on. When Sam says at the very end “Well, I’m back” he seems to mean that he is ready to resume his life and its story.

“Eucatastrophe” can reside in the way a story is constructed, so as to achieve that turn towards joy, or it can reside in the feelings aroused in the reader: preferably, the first leading to the second.

I think you are right to say that LOTR as a whole is not “eucatastrophic” but contains several good examples of the phenomenon. Helm’s Deep may be a very good example; Tom Shippey cites the events at the Field of Cormallen as pointing to “eucatastrophe.” His whole discussion is worth reading: it is scholarly but not, to me, convincing, based as it is on the fact that the Ring was destroyed on the 25th of March, which was the day (in tradition) of Christ’s conception, the Annunciation.

C. S. Lewis, being a much more blatantly Christian writer, may provide better examples. One that occurs to me is from the ending of The Voyage of the Dawn Treader, where the children seem to get a glimpse of Aslan’s realm:

It lasted only a second or so but what it brought them in that second none of those three children will ever forget. It brought both a smell and a sound, a musical sound. Edmund and Eustace would never talk about it afterwards. Lucy could only say, “It would break your heart.” “Why,” said I, “was it so sad?” “Sad! No,” said Lucy. (p. 212)

To answer the last part of your question, Tolkien may not have written The Lord of the Rings as a whole as a eucatastrophe because he was an artist, and an artist knows that a fictional tale, even a romance, has to have an ending. In other words his choice may have been based on literary and technical grounds. Or it may be an example of the old conflict between life and art, in which life has no shape, no chapters, no ending except one, whereas a work of art has to have shape and form, and also represent life.