Exporting a full WhatsApp chat to .txt
So, here's the solution I ended up using. Even though it's not ideal, it worked out, so I'll leave this here in case anyone else stumbles upon this.

The weak link in the chain is WhatsApp Web. Specifically, it's possible to write a tool that just scrapes the data WhatsApp Web receives, and someone on the Chrome Web Store did exactly that: The Backup WhatsApp Chats extension. It's like €3, so well within what I'd consider reasonable. However, I'd expect that anyone with more knowledge of coding than I have could write such a script themselves, because the messages in WhatsApp Web are apparently just displayed as plain HTML. 

It's reasonably fast and extracts all the non-media data you might want in csv format that can be parsed pretty well by pandas.

I'll accept my own answer because this is a solution within the parameters I set out, but if someone knows a better way, feel free to post it and I'll shift the accept.