If I've found the right site a question ... since you said 'public_html' folder ... are you on a shared host? or a dedicated host (VPS)?
One may not find php.ini in any app directories if on a VPS (dedicated) ... not needed. The main php.ini file is all that is needed then.
In a 2.6.3 version of Moodle and the config-dist.php file which shows some additional config items one could set:
[root@www moodle]# cat config-dist.php |grep temp
// results in the restore process stopping when a user attempts to restore a
// Keep the temporary directories used by backup and restore without being
// $CFG->keeptempdirectoriesonbackup = true;
// '/tempdir/' => '/var/www/moodle/temp', // for custom $CFG->tempdir locations
// It is possible to specify different cache and temp directories, use local fast filesystem
// $CFG->tempdir = '/var/www/moodle/temp'; // Files used during one HTTP request only.
Have any of the above set in your 2.4?
A 'warn' is just that ... a warning ... shouldn't stop the script in it's tracks.
Try backing up that troubled course, unchecking ANY *add-on* activities/mods/blocks ... you have to recognize them by how they were named and icons.
You should be able to manually remove anything found in moodledata/temp/backup/
Have command line access to your server?
If you do, could open a terminal session, and as root level user change into the moodledata/temp/backup/ directory. Then issue the following:
watch "ls -lR" [ENTER]
That command will show any activity in the directory - creation of files/folders, etc..
Then use your browser to begin a backup process. One of the first files created should be the 'roadmap' to the backup ... moodle_backup.xml. If that file is created and the backup process halts, that file might provide a clue.
Am taking stabs here cause I've not run into the issue exactly as described before.
'spirit of sharing', Ken