How to backup a secondary log shipped standby mode database
You could restore the same backup that was used to create the log shipped database on another server, and then apply all the transaction log backups.  There is no limit to the number of log-shipped databases you can have.  I don't know if Management Studio supports this option, but it is fairly straightforward process to implement manually (T-SQL, powershell, etc.) as you are simply copying files and restoring a database.  The database will need to stay read-only mode if you plan on restoring any more transaction logs.