This is the right answer: Break the fork link. I sometimes do this to make a private "fork" of a public repo so that I can add my own notes about how to use it, remind myself what happened when I tried it, add a config script for my own peculiar setup, etc.
It's unfortunate because not having a "real" fork makes it harder to send pull requests and track the upstream. But it's sometimes necessary to get around stupid github policies.
Don't you need to have an "GitHub-approved" fork (i.e. use the GitHub fork button) if you want to create pull requests on the upstream project in GitHub? Or is there a way to do that from the kind of repo you're describing?
Instead I prefer to use a "git" fork. I just clone it and upload it to my own repo. Assuming the license permits of course.