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.
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.