you should setup the file to check for the file permissions not the folder permissions.
So anyway... I looked at the file and had it echo the file name and there is no filename showing up so that is probably why it isn't working.
So I also looked at the post and added "sitemap-video.xml" to the root dir and changed the perms to both 0666 and 0777 and still not working becuase there is no filename being sent.
Your error message should show the file name you are trying to write. So people can manually upload a file and change the perms for the file only instead of the whole root folder.