Our team would like to copy several hundred test cases and retain the existing linked requirements. Since ALM treats the pasted tests as 'New' tests, is there a non-manual way to recreate these links? Thanks!

Answer
As you already know there is no way within the HPE ALM UI to copy defects with linkage from one project to another. Below is the HPE Knowledge Base article where this is dicussed.

https://softwaresupport.hpe.com/group/softwaresupport/search-result/-/facetsearch/document/KM870493

The only option that I could see that would allow you to retain this links would be to use the HPE QC/ALM API

http://alm-help.saas.hpe.com/en/12.20/online_help/Content/api_guides_main_page.htm

Another issue that you would need to be aware of is that there could also be defects attached to the tests, test instances, runs , run steps, and other defects.