I found the heroku fork -a PRODUCTION_APP_NAME NEW_STAGE_APP_NAME
to be a faster, easier way to do it... it creates the new app, copies everything (including the postgres database). Then I went in and manually downgraded the addons to smaller plans when it made sense (for example, a starter tier database).
In fact, we started using the relatively new heroku pipeline:promote
to manage automatically (and VERY quickly) pushing a compiled slug from staging to production. (That assumes you have any env-specific settings via settings or environment variables, so the code slug is the same.)