Release: Fix permissions for publish script

[JReinhold]

What I did

Use the a secret GH_TOKEN when publishing, to allow to push to next.

How to test

Checklist

• Make sure your changes are tested (stories and/or unit, integration, or end-to-end tests)
• Make sure to add/update documentation regarding your changes
• If you are deprecating/removing a feature, make sure to update
  MIGRATION.MD

Maintainers

• If this PR should be tested against many or all sandboxes,
  make sure to add the ci:merged or ci:daily GH label to it.
• Make sure this PR contains one of the labels below.

["cleanup", "BREAKING CHANGE", "feature request", "bug", "documentation", "maintenance", "dependencies", "other"]