I was having the problem where it would say everything was successful, and the server files were copied to the backup directory, but no .zip was created, and no database file was made. I found the problem to be that there was a large table (50k rows of data) in the same database as my WP tables, which it was trying to back up and caused a memory error.
I fixed it by adding a condition to the backup_and_move_backup_tables function:
foreach($tables as $table) {
//ONLY BACKUP THIS SITE’S TABLES!
if(strpos($table, $wpdb->base_prefix) !== false){
…
This ensures that only the table in the current site are backed up, and avoided my memory issue. Hope this helps someone!