I’d suggest using robots.txt - that should be able to support the feature you want (I think you can specify particular hostnames rather than just paths) and “good” search engines like google will ignore them.
Or, you can potentially use our custom headers feature to set a canonical link location, similar to what we do for the sitename.netlify.app requests automatically: Improved SEO with canonical link headers which by my understanding is also SEO-friendly.
Or, least easy, but surest it will work, you could make specific redirects for the “wrong” hostnames:
https://thing.oursite.com/* https://thing.com/:splat 301!
…to make sure the crawler starts indexing in the right location.