[BelleNottelling]

Unfortunately the spamchecker settings aren't being saved in the database, nor the client signup settings.

Oh I see.
If memory serves me correctly, the last time I saw this the culprit was an overly aggressive caching layer that was placed between the FOSSBilling instance and the browser.
The updated data was indeed being written, however the aggressive caching meant the previous page content was being sent to the browser, resulting in the visual appearance that nothing was actually being updated.

What's your server setup?
If you check the network tab in your browser, does it give any indication that the page contents were cached?

As for which table it's being written to.. IIRC it's the extension_meta table, however the configurations written there get encrypted so it's not really something you can just go in and edit.