You're already saying that you're taking actions based on what legitimate criticism you find on the hate site, but without acknowledging the source. Given this, the only question is whether you'll use the specific moderators targeted by the site in order to leverage the hate site into advertisement for your own site - which strikes me as a very appropriate use of hate (and a nice example of "engineer hoist with their own petard"...)
The one downside to this that I can see is that there is a risk of the haters upping their game. In other words, they may not stick with hating on their own site, and start actual harassment of the mods they dislike.
So if you do this, it should be with the informed consent of the moderators in question, and you should be ready to help and support them if something happens.