Map client to alternate virtual directory based on url prefix?

I have an app that has an external url that is different from the internal url.  Lets refer to them as:

https://external.app.com/virtualdir/app.exe   (uses to forms based authentication)

and

http://internal.app.com/virtualdir/app.exe   (uses integrated authentication)

Currently the external url is being served up via a lightweight proxy which we plan to eliminate in favor of a direct connection via an F5 device.  

However, on the iis7 webserver (server 2008), /virtualdir is set to integrated authentication.  Is there a way to detect that the client is using "external.app.com" and map them to the forms based virtual directory,  /virtualdirforms for example, all while keeping the /virtualdir url syntax on the client?  

I know I can create a new website on the server and create another /virtualdir with forms based authentication set, but from what I gather, it would require another IP to be allocated to the server, which would require I jump through a bunch of red tape that I don't want to do.

Hopefully this makes sense, I can clarify if need be.