>The app lets the user feed the book into an LLM, as is absolutely their right,

I don't think that's cut and clear yet. Throwing media onto someone else's server may count as distribution.

How likely do you think it is that Amazon doesn’t have a pre-existing contract with these publishers to host these books on Amazon servers?

Sure, in the sense that any belief about the law isn’t cut and dried until a judge has explicitly dismissed it in the court of law.