I think I get it:
chagrinfalls.net is where the content lives
chagrinfalls.dreamhosters.com is a mirror of chagrinfalls.net so you can test it out.
Now the client wants to use example.com instead of chagrinfalls.net.
The easy way is to Mirror so that chagrinfalls.net shows up as example.com
The right way is to create a Fully Hosted example.com as the same FTP user as chagrinfalls.net. This will create an example.com folder next to the chagrinfalls.net folder. To finish up, you'll have to update any references to chagrinfalls.net so they're now example.com.
Rename the example.com folder to example.old, and then rename the chagrinfalls.net folder as example.com. For completeness, you can rename the example.com folder as chagrinfalls.net, but you'll probably be deleting it anyhow, and maybe even make chagrinfalls.net a Redirect (not a mirror) to example.com.
This will break the mirror, so you can create a new one as example.dreamhosters.com, or delete the chagrinfalls.dreamhosters.com and then make it a Redirect to example.com.
The right way is a bunch of work, but if you're going to settle on a domain name, then that's where everything should be hosted. Mirrors aren't search engine kosher, as you're disguising the true source, which is why I suggest a Redirect.