Is there any animal internal monologue in The Lord of the Rings? Does Tolkien ever include any internal monologue from the perspective of an animal? That is to say, a scene where he shows an animal talking to itself inside of its own head?

Yes. When leaving the Shire, the hobbits disturbed a fox:

> A fox passing through the wood on business of his own stopped several minutes and sniffed.
>
> 'Hobbits!' he thought. 'Well, what next? I have heard of strange doings in this land, but I have seldom heard of a hobbit sleeping out of doors under a tree. Three of them! There's something mighty queer behind this.' He was quite right, but he never found out any more about it.
>
> \-- _The Fellowship of the Ring_ , Chapter 3: "Three is Company"