Thank you for the prompt reply, Falko.
Would you consider this to be a valid "feature request"? Due to the fact that ISPConfig forbids this configuration, I now have to:
1.) Move the site in question to a completely different physical server (even though I have plenty of unused IP addresses on the current server). This is a major headache because we don't allow remote MySQL connections, so a second database has to be stood-up on the new server, and the data in each DB merged later on. (Replication would be ideal, but is outside the scope of this change.)
2.) Update the DNS records to point to the other server.
3.) Once look-ups are resolving to the new IP address (on the separate physical box), change the domain in the Website tab on the original server.
4.) Update DNS to point back to the original server (but different IP address).
5.) Delete the "temp" site.
It seems that need to move a website from one IP address to another, on the same server, should be fairly common, especially given the lack of ubiquitous support for SNI at this time.