Is this a supported scenario in LMS?
I know of the possibility that you can backup the site collection, delete it and restore it under a different URL. Is this supported?
Yes, this is a supported scenario in LMS but it'll require one additional step to connect the newly created course to the organization site.
Steps would be as described below
1. create a course backup using stsadm -backup -url http://courseurl -filename C:\temp\course.bak
2. restore the course using stsadm -restore -url http://courseurl -filename C:\temp\course.bak
3. run SPLMSUtil.exe -o attachcourse -organizationurl -courseurl
The tool you will find at http://help.elearningforce.com/Downloads/SharePoint%20LMS/LMS%20Tools/RestoreOgranization.AttachCourse2013.zip
4. Make sure everything worked fine
5. delete the course with the old URL from organization
PS Please keep in mind that standard sharepoint backup-restore does not convert the absolute links inserted into site content i.e. it will not point to new URL in case if it was inserted like http://oldcourseurl/link