Private Github Plugin with Authentication Token

I’ve got a plugin which I’m developing. I’ve got the source code on a Github Repo and I’m checking for releases. This repo is private which is fine for checking for an update because I can set the Authentication code in the header using CURL but I don’t know how to set the Authentication code in the header for the update button:

I know Github was previously authenticated using URL parameters but they’re no longer supported so I need to set it within the HTTP Header.

I’ve tracked the update button to the plugins_api filter. Is there another filter I need to add where I can set the Authentication token or is this not possible?

If not how do I make it known that the new version needs downloading and installing manually?

Please let me know if you need any more information. Thank you in advance