Just wanted to follow up with some our findings and how we resolved it, in case anyone else finds themselves in the same situation.
With help from Hook and username, we discovered that it wasn't a simple cname. They were scraping the site and creating a cached copy on their server, whilst also slightly modifying it.
New pages were loaded on demand and old ones refreshed once the cache expired (the fact that the new pages were showing immediately was one of the things that made me initially think it wasn't a simple scrape job).
There was also some javascript added to show adverts for an asian casino, although they weren't showing for us, so I'm guessing they were limited to specific types of user. I'm not really sure where they were planning on getting traffic from, but at least now we know the reason behind it.
We located their IP by trying to load a URL that doesn't on our site. This then showed up as a very obvious entry in our server logs (eg: "/here-is-the-ip-of-the-bot/") so I blocked the IP and new pages immediately stopped loading on the other site.
It took a while for the cache to expire, but now it has, the entire site is down. So problem (temporarily) solved.
Obviously they could just change the IP and start again, but for now their site isn't working, at least.