Adam talked me through this. Short answer: in the current version of enhance you can change the IP addresses easily, so I changed the affected server's IP address to its local (non-routable) address, did the same with the panel, migrated my web sites off the server, changed the IP addresses back to the routable IP addresses, and now (at least with my first trial) web sites can migrate cleanly between both sites.
It looks like my problem was solved without having to rebuild the server.
