Run GitHub publishing steps during unofficial pipelines

[lbussell]

This is a follow-up to https://github.com/dotnet/dotnet-docker-internal/issues/8105.

With the introduction of official/unofficial pipelines split, we'd like to exercise publish steps that reach out to GitHub during unofficial builds, if possible.

This includes:

• Publishing image info:

  https://github.com/dotnet/docker-tools/blob/7841dec2366245a4446a48617e7fbf68b27fa98e/eng/common/templates/jobs/publish.yml#L242-L255

• Build notifications:

  https://github.com/dotnet/docker-tools/blob/7841dec2366245a4446a48617e7fbf68b27fa98e/eng/common/templates/jobs/publish.yml#L257-L292

If possible, we should have a separate GitHub identity/App which has access to an internal/non-public repo that we can push these changes to.