Ah, I found the problem. I had originally purchased the cMap template while I was working on this in development. I didn’t realize that the license was tied to the development site. So that’s why the plugin wouldn’t update. It’s also why when I deactivated the license and tried activating it, it wouldn’t work.
So I went to my account, manage site and deactivated the old development site, then put in the URL of the production site. Went back and activated the license and it worked. Now the plugin updates successfully.