To answer my own question: This is not possible without rebroadcasting the streams. .m3u8 files are files containing links to other files, which in the end also contain the binaries. All of these, including the HTTP response containing the binary, needs the CORS headers for the Chromecast to display the contents.
If you're only looking to add CORS headers to textual responses corsproxy.com is a good alternative, a long with several available open source projects.