With some extra careful prompting, you could probably get a GPT model to write lyrics that really would read like they belonged on Castaways; you just need more archaisms. You probably could take these prompts and just tell ChatGPT "ok, but replace some of these words with archaic almost-synonyms that sounds like they mean the same thing but really don't" and get 98% of the way there.
This Slate article does a decent job of capturing some of what I mean:
https://slate.com/culture/2009/04/the-eight-most-pretentious...
You might, like me, read this and get these sense that Meloy's lyrical style is probably very trainable/promptable.
PS
I get that I'm talking exclusively about the lyrics, which are of course the easy part of songwriting for ChatGPT. I just think this is a funny particular comparison to draw, this particular band. Which, again, I do love.