Hi David,
Well, if you use a tool that can change the name of the bot that comes after a site, then you can't do anything about it. However, I do know that SEMrush is causing problems for some hosts and the likes of Heart Internet, they block access to the SEMrush bot on all of their servers.
To be honest with you, it is quite difficult to block everything and time consuming and unless you have some secret formula to keywords that are magically drawing in more traffic, then I wouldn't worry about it too much.
Take Accuranker. I use this for tracking and part of setting up, is it will find new phrases that I perhaps haven't been using or looking at before. It doesn't get these from either SEMrush or Ahrefs. All of this information is available from many sources.
I would honestly spend the time making your pages as good as they can be. Look at your cometitors. Are they winning? How are they doing it? Fight back against them rather trying to hide yourself from view.
-Andy