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.
3If you can find those moments from Shelob, quoting them would improve this answer. (Although I'm not sure if a superintelligent supermassive monster really counts as a normal enough animal to fit the scope of the question.)– Rand al'Thor ♦Dec 30, 2019 at 6:12