Hi Catalyx,
I did spend about 30 minutes having a look, and my only conclusion is that this is extremely bizarre. The data that it says (from yesterday's message) does not exist is indeed in the database. And this is code that is run on every UD installation, every time a backup runs. It's code that should work always or never. But this is apparently the only site where it is dodgy.
I wonder if it will work tomorrow. I am wondering if the difference between "working" and "not working" is that someone is logged in to the WordPress dashboard at the time. I can't think how that can be, but it's the only difference between 'working' and 'not working' that I can see at the moment.
If it fails tomorrow, then the next day turn off WP Better Security and see if that is doing something that interferes.
David