Is it possible to fork a public GitHub repo into an enterprise repository?

It's not possible, because your GitHub Enterprise (GHE) installation is separate from the public one, so there's no way for the two systems to track each other's branch relationships.

The best you can do is exactly as you describe: Clone the repo, then push it to your GHE installation, and yes, you will then lose the branching relationships across repos.

The other option would be to keep a fork on the public GitHub (GH) repo--possibly keeping it in sync (manually, or with a cronjob) with your GHE repo. Then you'll have two instances of your repo, and the public one would retain branch relationships with the original repo.

Depending on why you need to put this on GHE, it may or may not work. If you're making private contributions, it clearly won't work--as your private contributions would no longer be private. If you want it on GHE due to some corporate policy that all open source projects used internally are kept on the GHE, or something similar, then it would work, with the added administrative overhead of keeping the repo in sync two places.

While it isn't possible to fork from the public GitHub directly to your Enterprise installation, you can fork it on the public GitHub and then mirror that forked repository on your enterprise installation.

Steps

1. Create an empty repository on your enterprise GitHub:

   curl https://github.yourenterprise.com/api/v3/user/repos \
     -u "yourUsername" \
     -d "{\"name\": \"whatever-repository\", \"private\": true }"
   

2. Create a bare clone of your fork

   git clone --bare https://github.com/publicGitHubUser/forked-repository.git
   

3. Change directories so you are inside the bare clones folder:

   cd ./whatever-repository.git/
   

4. Push the repository with the --mirror flag to your enterprise GitHub

   git push --mirror https://github.yourenterprise.com/enterpriseGitHubUser/forked-repository.git
   

More Information

https://help.github.com/articles/duplicating-a-repository/

If you use GitHub Enterprise then you automatically get access to GitHub Enterprise Cloud i.e. https://github.com/enterprises/mycompany, with the same licence count. This can be linked with SAML to your in-house Active Directory. Then you can simply fork it in GitHub yet keep it within your company's control.

Sadly you will still need to mirror clone it if you really need it on your in-house instance. However as you control both ends now, you should be able to use GitHub Actions (or Apps) to more intelligently keep them in sync.