How do you bulk copy several test suites, test cases, test scripts within RQM?

There is a "duplicate" capability that can be invoked either on an individual object or on multiple objects selected in a list.
https://jazz.net/help-dev/clm/topic/com.ibm.rational.test.qm.doc/topics/t_copy_asset_while_viewing.html

Duplicate will allow you to select a different project area as the target of the operation, assuming you have write permissions to that project area.

Have you tried Rational Quality Manager Copy Utility ?  This is available in download section.

Check the spec of this utility 
  https://jazz.net/wiki/bin/view/Main/RQMCopyUtility
Also recommend to test in a test environment ( not production ! ) as this may not be what you want.

Dennis is correct, You can copy artifacts within project area provided you have permission to write in target project.

Duplicate option would help you choose the project area hosted on the same server provided you are a member and have write permissions to those project areas.

In-case you are trying to transfer the artifactsto a project area hosted on a different server, you will require Copy Utility.
This utility can be downloaded from jazz.net under All downloads.

Please note that the copy utility is compatible between the same versions of the server.

Hope this information helps!!

Best Regards
Rajat

Is RQMCopyUtil the only way to get test artifacts from an RMQ project in server1 to another RQM project on server2?  Is there any other option?  When I try to use the utility it fails each time.