Add an alternative IP address to the network card on the host, and use this IP address for the web server. Alternatively, you can set the IP addres in the virtual machine and use that.
As another alternative - have a sys admin update your DNS server; add aliases for your web server and set up your headers to handle these requests, assuming you're using IIS. I'm not sure hw to do this on other web servers.
To update DNS settings, on a domain controller go to the DNS management console and browse to the appropriate domain in the forward lookup zone. Add a new alias and point it at the web server. The alias may be something like MyWebApp.MyDomain.Local, and point at MyWebServer.MyDomain.Local.
If this sounds like a suitable way to do things, let me know and I'll add some more detail on how to do this.
Which might not be answering the question asked... :-\