I have set this up once, so I know it's possible, just do not have the script previously used to replicate the process:
Step 1) Execute the Backup of DatabaseLive (DatabaseLive.mdf & DatabaseLive.ldf) to DatabaseLive-MM-DD-YYYY.bak
Step 2) Restore the DatabaseLive-MM-DD-YYYY.bak to DatabaseTest (DatabaseTest.mdf & DatabaseLive.ldf)
Step 3) During the restore the Database and Log File need to be changed in the Restore to location so it does not overwrite the live database and log files.
Step 4) Replace DatabaseTest table sysconfig field orgname (sysconfig.orgname) with 'Orginization Test -' + Current Date
It took me awhile to get it setup previously, but that was awhile ago and I have since left that position.
Any assistance would be greatly appreciated.