That is very good information! Indeed your PB is having issues writing to the filesystem.
The way updates work is they first install the recovery.html app, then replace (deleting) the main app with a copy of the recovery app so that if the next step fails, you have an easier time fixing things. It does this to free up space, and if something happens it can leave you without the main app.
If you can, try to use @pixie’s tool to get a backup. Otherwise, files can be downloaded manually by URL.
You can get a file list with the /list endpoint. This just gives filenames though. There may be a few extra files that aren’t critical that can be deleted, like apple-touch-icon.png and favicon.ico. Most of your storage is going to be used by patterns though.
Files can be deleted by issuing a request to /delete?path=/somefilename.
Files can be uploaded too via a POST with something like curl, but that is where using @pixie’s tool would save you a lot of work. This post has a few more details, but was written before this tool was created.