'public_html' even on a dedicated server is in a 'user jail' ... trust me when I say that!
So you do have ssh into server and can call up the mysql client app on the server.
So again;
mysql> show variables like 'max_allowed_packet';
what does the above show.
Assigning globals via command last only as long as the DB server runs that session. When/if DB server is rebooted or restarted, that global setting is lost. To be permanent, has to be in config file on start up of DB server.
You could also do a full backup of that course from the command line ... takes apache out of the loop leaving only php and mysql server then.
cd /path/to/your/moodle/code/admin/cli/
php backup.php --courseid=[idnumberofcourse] --destination=/path/to/where/you'd/like/to/save/it
Let's try that to see if it errors or finishes ... but only after making sure max_allowed_packet is set the way you think it is.
If it does error, the command line backup.php execution should show errors ... come back here an post the text of that error.
'spirit of sharing', Ken