Here's a picture of that wire. It's just wrapped around the posts at each end. It reminds me a lot of a fusable link-type of wire. I have good reason to believe that this regulator is good, save for this one wire that is burned through. I'm sure that it has a specific value for the right amount of protection so I don't want to "put a gum wrapper around it" so to speak.
The newly adjusted regulator is working fine. Hopefully, continues to do so. But, if it should ever decide to take a dump, I'll hopefully have this one ready as its replacement.