Should have done more testing, I guess.
Leaving the FTP layer enabled, but *manually* setting the 777 permissions to the appropriate directories in the master and ALL slave sites seems to have fixed it.
Apparently there is just some item that does not get accessed appropriately by FTP, but having full permissions fixes it.
Since there were no error logs, FTP logs, or access logs with the specific problem directory, it's difficult to know what exactly fixed the issue.
But I guess we know that at least with FTP layer enabled, directory permissions (as listed in the Joomla System Info page) must still all be manually set to writeable outside of FTP.
Thanks!!!