How to use `click` in a shortcut? (Run a service when clicking a link)

Is it possible to run an Automator service with an alternate click, e.g. ⌘ + ⇧ + Click? This kind of combination is built into browsers with ⌘ + Click typically opening a link in a new tab. I have a service for handling links. Right now I can right click the link, pick services from the context menu, and then select my service. I'd rather not navigate the menu. Is there a way I can harness Click when making a shortcut? A half-option is to right click the link, click elsewhere to close the dialog, and then press the keyboard shortcut, but this is cumbersome. (I'd be happy enough if I had to right click and then do a shortcut, but this seems to not work without clicking away.) I've internet sleuthed with enough search strategies that I think this may not be possible but am hoping I'm wrong and simply a careless querier!