Thanks to everyone who replied! In the end, @Graphus was right. It turns out that sometimes all you need is some WD-40 and good old brute force. The fastener came loose after a bit of convincing with a hammer, and thankfully the wood was not split or damaged further. Success!
Use your favourite web client to search for terms like: click-clack hinge, sofa bed mechanism, etc.
There are many variations on these, but the idea is that they are a hinged locking mechanism with two (or more) resting positions.
Whether or not you can find a set that match your aesthetics is up to you. There are some that are meant to be exposed.
You say that 3 sides fold up, but you didn't show the detail of how the end/side interface works. However, I'm thinking that some sort of decorative hook & clasp might work to hold them in the "up" position.
Something like this:
Image courtesy of Lowes.com. No recommendation of vendor or individual product implied or intended. For search term ...
I'll collate the info from the Comments above into an Answer on what's causing this and what to do moving forward.
What is causing all these cracks and is it likely to happen more?
These look like classic, textbook in fact, drying cracks. As they've occurred after the wood was supposedly dried it seems likely it was poorly done, either too quickly (leaves ...