I verified that the problem is not permissions related. In fact, it appears that my issue may in fact be a 'BUG' in Moodle version 1.6.1.
My attempts to resolve this issue led me to placing a trolley load of debugging code in /backup/backuplib.php and other script files. I found that if I added a 'Display a directory' resource to my course and set the Display a directory setting to 'Main files directory', then this caused the course backup to fail.
I further discovered that the backup process treats the Display a directory resource type as a regular file directory, and copies any subfolders and files in it as part of the backup process. From what I can figure out, this is in fact the problem, because when the backup process reaches the 'Copy Course Files' phase, it is attempting to copy files that have already been included in the backup, and is therefore failing somewhere along the way. I verified this by removing the extraneous 'Display a directory' resource from my course, and the backup completed successfully!
To contradict my findings, I tested a similar scenario on a 1.6.1 Moodle installation on my laptop. In this case, the backup worked regardless of whether or not my course contained a 'Display a directory' resource.