You want to re-deploy the same commit SHA from a private GitHub repo.
This can be done with the builds-create Platform API endpoint.
There is a Builds CLI plugin that makes this easier:
heroku builds:create --source-url https://user:firstname.lastname@example.org/repos/<username>/<repo name>/tarball/master/ --app <app-name>
user is your Github username and
token is a Github personal access token (reference)