I'm not sure this is much more likely as it has some of the same problems, but Tolkien's poem The Adventures of Tom Bombadil did exist in print at that time, in a 1934 edition of the Oxford Magazine, and I think better fits the description "fairy story".
Incidentally, I have the 1976 printing of The Father Christmas Letters, and the 1937 entry is different. In it, FC explains he's strained his hand and most of the letter is a "picture diary" by Ilbereth. Neither mentions Hobbits or a different fairy story. I understand that the later version "Letters From Father Christmas" is different from the earlier one in some respects, but how can there be two different letters for the same year?
Incidentally, I have the 1976 printing of The Father Christmas Letters, and the 1937 entry is different. In it, FC explains he's strained his hand and most of the letter is a "picture diary" by Ilbereth. Neither mentions Hobbits or a different fairy story. I understand that the later version "Letters From Father Christmas" is different from the earlier one in some respects, but how can there be two different letters for the same year?