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"
If I remember correctly, Shelob the sentient spider has a few inner-monologue moments.
Also, if you're interested about regular animals showing sentience in general, in The Hobbit, a Thrush and a Raven communicate with the Dwarves when on the Lonely Mountain. The Raven, if I recall correctly, tried to communicate and wasn't understood by the dwarves, but understood the dwarves' speaking about needing a Thrush, and the Thrush spoke with one of the dwarves who could interpret.