I think it needs to be evaluated relative to the volume of the work. If you take a novel in the public domain and just change a few sentences, I don’t think you can claim copyright for the modified novel.
The courts are going to always be obsessed with one practical scenario. I take e.g. Harry Potter, put it through an LLM with the prompting of essentially writing the exact same thing with slightly different characters, locations, and phrasings. And then put it up for sale (or free download), making no secret of the fact that it's essentially the same book.
If an LLM reaches a sufficient capability that this can be achieved while maintaining a roughly comparable level of readability, then you'd have just completely legalized 'piracy by proxy'. And the same would come to every single medium from movies to software. This is going to make it an extremely difficult question to answer.
I expect what it come down to is determinism. LLMs are completely deterministic - same model + same input + same seed = same output. And so any seller will be obligated to retain any model composition/data, and any copyright holder of an input training item will have a copyright claim on any output item. In other words, you can only train on stuff you already have the copyright to, or rights to.