Finishing up the section in which Tolkien discusses the types of stories that he doesn't deem "fairy stories..."
There's not a great deal to add with respect to Tolkien's exclusion of "any story that uses the machinery of Dream, the dreaming of actual human sleep, to explain the apparent occurrence of its marvels." The only example he gives is that of Lewis Carroll's two ALICE books, which probably were deemed "fairy tales" by some adults simply because they contained marvels (albeit rationalized) and were (in theory) aimed at children. But as Tolkien says in his footnotes, it's also widely recognized that the ALICE books are satire, which carries little mood of "faerie."
At the same time, there do exist works that evoke a strong "faerie" touch despite resorting to dream-mechanism. Ironically, though the Oz books of Frank Baum are not justified as dreams, MGM's 1939 film does convey a strong mood of enchantment, despite telegraphing the "only a dream" ending early in the narrative.
Tolkien's ruminations on the "beast-fable" are a little more involved. It's true that there are many stories-- particularly those of Aesop and Uncle Remus-- in which the only marvelous element is that animals are portrayed as doing very human things. And Tolkien is also correct in saying that such stories, insofar as they cease to deal with "the proper languages of birds and beasts and trees," do not possess the faerie charm; that they are merely extensions of a human ethos and do not address the faerie desire to know "communion with other living things."
Still, Tolkien provides insufficient examples of those stories that successfully blend faerie magic with anthropomorphic animals. One might expect him to discuss traditional folktales, since he has been so insistent about disincluding literary fantasies of various types. Instead his positive examples include "the stories of Beatrix Potter" and a George McDonald tale. He mentions the Egyptian "Tale of Two Brothers" as a positive example of the motif of the "soul/heart separated from its body," but this doesn't support his point about beast-fables. He seems to have tossed it in as a counter-example to the beast-fable "The Monkey's Heart," in which a character alludes to such a separation but no such separation actually takes place. In my somewhat humble opinion he might have done better to have discussed a traditional folktale in which the beast-element is tied in with some magical circumstances, such as-- just to name one at random-- the Grimm's tale Hans My Hedgehog. In contrast to an over-anthropomorphized "fairy tale" like "Puss in Boots," "Hedgehog" should satisfy Tolkien's desire to find an "arresting strangeness" in his tales of faerie.