In TextMate, there is an option to import external files, but it only creates references to those files. How can I copy the files rather than reference them?
TextMate projects and its groups aren't locations where you can copy files to. Note that the groups' "folder" icon have a different color than regular folders.
You need to add an actual folder to your TextMate project e.g. by dragging and dropping it onto the project files and folders list from Finder. Then you can copy files you want to be part of the project to that folder (use Finder to do that).