I went ahead and took the plunge anyway and tried to do it myself, manually. What I did:
Rename the /var/www/web15 directory to /var/www/web6.
rm the www.example.com
symlink in the /var/www directory (it's broken now that the web15 directory no longer exists). Recreate the same symlink in the /var/www directory using ln -s to point to the new /var/www/web6 directory.
Rename the /var/www/web6/user/web15_www.example.com directory to /var/www/web6/user/web6_www.example.com
Renaming the directory also broke the /var/www/web15/Maildir symlink. Delete the Maildir symlink and recreate it using ln -s in the /var/www/web6 directory, pointing it to /var/www/web6/user/web15_www.example.com/
Edit /etc/apache2/vhosts/Vhosts_ispconfig.conf file and change all references for the old /var/www/web15 directory to /var/www/web6 in the virtual host config for the specific virtual host I wanted to change (www.example.com)
Restart apache2: /etc/init.d/apache2 restart
The website appears to be working the same as it was before all the renaming (though it was mostly broken already because it needs php4 and not php5 to run properly, but that's another issue).
Being a relative newb, I have very little idea of whether anything I did was the right thing, but it seems to have worked. The site still shows up in ISPConfig, too. CAUTION: DON'T ATTEMPT TO DO THIS UNLESS SOMEONE WHO KNOWS BETTER CHIMES IN AND SAYS THAT WHAT I DID WAS OKAY. No telling what I screwed up on my test server messing around with it like this.