If they are scraping, there has to be some evidence in the server logs. Since you can tell from "snapshots" of their server of when they last updated, it shouldn't be too hard to see an IP address, or range of addresses, is accessing this site, at and just before the most recent times shown.
I had one of my sites mirrored about a decade ago, and back then it was no problem at all to find the offending server and block it completely. Things are more easily concealed now, but still, there are ways to find out the IPs scraping the site regularly. I've had to do detective work in the past to find out why our dedicated server, which more than supported our number of users, was unresponsive at certain times. Back then, I discovered it was baidu-bot, from the Chinese search engine Baidu, slamming us hard with multiple requests per second. (And with each request hitting the database to display a thread, well...you can imagine why it was slowing things down.)
Good luck in the search. Getting them shut down may be a chore (throwing the terms "copyright" and "trademark" around if needed, and if applicable), but finding the "leak" through which they are scraping content and mirroring the images can help. Even if it's a lot more work.