SQL Server Data Refresh
Would it be possible to mount additional drives on the target SQL server, restore the data files to the new drives and then simply change the location of the data files in the DB once the restore is complete?

Yes, if the two databases on the target server have the same file layout.  A simpler and safer procedure is to restore the database with a different name, and then rename the databases to swap.
If you can avoid detaching a databse, setting it offline or shutting down SQL Server you have extra protection against losing data.