What's the equivalent of `heroku container:release web` without using heroku-cli?
You can install the Heroku CLI via npm - which might be the simplest. 

Note that your Heroku API key has to have the env var name HEROKU_API_KEY for the Heroku CLI to accept it as an auth token (ie. to avoid having to do heroku login). 

          - npm install -g heroku

          # build the Docker image (this will use the Dockerfile in the root of the repo)
          - docker build -t $HEROKU_APPLICATION_NAME ./backend
          # authenticate with the Docker Hub registry
          - docker login --username=_ --password $HEROKU_API_KEY registry.heroku.com

          - docker tag $HEROKU_APPLICATION_NAME registry.heroku.com/$HEROKU_APPLICATION_NAME/web
          - docker push registry.heroku.com/$HEROKU_APPLICATION_NAME/web
          - heroku container:release web -a=$HEROKU_APPLICATION_NAME