Hi Shyfty,
I apologize for the slow response. I was doing some checking to try to find the simplest solution for you.
It looks like what you are going to have to do is make a local copy of your fork of the Engine Source from Epic, then create your own repository based on your local copy (using GitHub, or whatever source control you choose to use). You can then make your repository private and allow any of your team members to access it.
In order to keep the Engine you are using in your repository up to date with the latest release version Epic has available on GitHub, you would need to sync your original fork to the latest updates from Epic, then merge those changes into your local repository.
One final point, which you may already be aware of but I wanted to mention for clarity and future readers of this post: Each member of your team that will be working with your local repository to access the Engine would have to have a subscription for Unreal Engine 4.
Please let me know if you have any additional questions.