Is there a way to import a csv list of new records related to another module by checking/matching a field or group of fields other than the ID?
I think (not sure) that in some cases, yes. For example, I seem to remember using an Account name when Importing Contacts or Leads will link to the correct Account, if it exists. Maybe you can try this to see if it works.
But I doubt it works in the general case, the importer wouldn’t know what to do (which field to match on the related table). It will probably require code customization.
There are some importing scripts here on the forums, maybe you can adapt one to your needs.