@David
Even if you delete files contained in a course and linked to in that course in the Moodle UI, it may not help. Moodle 2 doesn't tokenize and create records in the database that represent the files. It does, however, have metadata that pertain to them (humanly recognizable file name, size of file, mime type, etc.). The actual physical file's name is hashed and placed in filesdir of the data folder.
The table Moodle is trying to write to: mdl_backup_controllers. That line that you edited out: 12 => VERY LONG STRING! adds to the issue by may not be the culprit by itself ... depending upon how Moodle attempts to write to mdl_backup_controllers. The database will not accept the total chunk of data Moodle is attempting to send to it.
If you are remotely hosted on a shared server, you may NOT be able to add any my.cnf file to control the database server ... which is also shared by multiple customers. Most setups like that *are* restricted ... that's why they are so cheap! ** You must contact your provider and inquire about increasing max_packets for your setup. **
'spirit of sharing', Ken