In seriousness, probably not, unless US "intellectual property" law gets worse somehow.
Short phrases fall under trademarks rather than copyrights, and even then it needs to be something that would cause commercial confusion, and very few people are going to buy a Tolkien book expecting a nuclear reactor or vice-versa.
I thought that was mostly about coincidentally similar names.
Does it change when one is explicitly a reference to the copyrighted work? It's not like Thiel just thinks that Valar and Palantir and Anduril etc are all just nice sounding words, he's built a brand out of companies named after Tolkien's stuff.
I happen to know for a fact that Palantir licenses the name from the estate at an exorbitant yearly fee.
IANAL but I would assume that such an agreement (A) falls under trademark law and (B) is being done to avoid a costly/embarrassing fight, even if it's a fight Thiel might win.