Update WordPress Theme / Plugin from Private GitHub Repo

Background

I am working on a custom theme for my WordPress site which I would like to manage from a private GitHub repo. (This theme will never be pushed into the WordPress market place) The general idea would be that I use the repo to manage the code and then once I tag a new version, the tag would trigger an update for the WordPress theme.

I have this pattern working using the following as a template:
https://github.com/krafit/wp-gitlab-updater
(Yes, I know the repo is for Gitlab and not GitHub)

Since my repo is private, I will need to generate a user token to allow the theme to be updated. And because the user token is capable of accessing all my private repos, the idea of sharing the user token with another plugin is discomforting from a security standpoint. (Meaning, I'm uncomfortable using a plugin like: https://github.com/afragen/git-updater)

Question

The problem is that GitHub has deprecated the use of access_token as a query string parameter, so all tokens must be sent over as an Authorization header.

How do I add an authorization header to the request WordPress sends to download the artifact?

What I've Tried

When I check for new tags I use the code:

  protected function fetch_tags_from_repo( $git_url, $repo, $access_token ) {
    $request_url = "$git_url/repos/$repo/tags?access_token=$access_token";
    $args     = [
      "headers" => [
        "Accept" => "application/vnd.github.v3+json",
        "Authorization" => "token " . $access_token
      ]
    ];
    $request     = wp_safe_remote_get( $request_url, $args );

    return $request;
  }

This works without any issues. However...

During the pre_set_site_transient_update_themes hook I return an object that looks like:

  $transient->response[ $theme['name'] ]['theme']       = $theme['name'];
  $transient->response[ $theme['name'] ]['new_version'] = $latest_version;
  $transient->response[ $theme['name'] ]['package']     = $theme_package;

The problem is, I have no way of adding an Authorization header to the transient response object. Therefore, when WP later tries to download the artifact, it fails.

Note: The $theme_package string is a URL which looks like:

$theme_package = "$git_url/repos/$repo/zipball/refs/tags/$latest_version";

Any support appreciated, thank you!

Eject from GitHub and use Gitlab because they still support access_token as a header. They have unlimited free private repos <5gb storage.