I'm curious if they are stealing anything else, e.g. are they selling ads/tracking, do they replace order forms with their own...
Also to teach them an important lesson about the internet.
there was another time a site hotlinked to a js file. After asking them to stop, i found that they had a contact form with a homebrew captcha which created the letters image like http://evilsite.com/cgi-bin/captcha.jpg?q=ansr
A little while later, their captcha form had a hidden input appended with the correct answer value, and the word to solve was changed to a new 4 letter word from a dictionary of interesting 4 letter words. The form still worked because of the hidden input. I might have changed the name on the "real" input also.
Additionally if they decide to blackhole the fake/honeypot url, since you mentioned they pass along the user agent, you could mixin some token in a randomized user agent string that your scraper uses so that you could duck-type the request on your end to signal when to capture the egress ip.
Reminds me of a time some real estate website hotlinked a ton of images from my website. After I asked them to stop and they ignored me I added an nginx rewrite rule to send them a bunch of pictures of houses that were on fire.
For some reason they stopped using my website as their image host after that.