Another option would be to use a service like Cloudflare, which offers protections against scraping and other malicious behavior. This can help prevent the proxy-mirror site from being able to access your site's content.
Just an idea but maybe you could cause big loads on their servers by requesting in parallel a large amount of urls where you actively serve a gzipped massive html file that is full of links to your website.
EDIT: or building up on what user zhouyisu says above you can generate your perfect match IP blacklist by calling urls via the abusing site that automatically puts any caller into the blacklist.
https://blog.cloudflare.com/introducing-scrapeshield-discove...