How to copy a database in sharded cluster
If you're using 3.2 and above you can use mongodump (Docs found here) and mongorestore (Docs found here).  Note that the docs state it can be problematic to backup sharded databases.  You will have to consider how in sync you want the data and take proper action.  Mongo docs going deeper on that can be found here.

Mongo 3.2 supports backing up sharded clusters with a snapshot, if you want to use that option check out the docs found here.

Finally, restoring the databases using the various processes can be found on Mongo's docs on restoring sharded clusters found here.  Again this will differ based on what version you're running, so be careful to read the docs.  They all go through the process in depth and clearly.

I thought you could use copyDatabase, but the docs state that it does not work with sharded databases.