Trying to open an IIS page locally

I have two IIS sites, which expose the same virtual directory.

One is bound as http::80:wfe.website.com, and is accessed through a reverse proxy server.
The second is bound as http::80:www.website.com, and is not exposed externally.
Our public DNS entries for this website refer www.website.com, and point towards our reverse proxy servers.

We have an entry in our host files, pointing www.website.com to 127.0.0.1.
I have no proxy settings in my LAN settings.
If I ping www.website.com I correctly go to 127.0.0.1.

Now for my issue: On 5/6 of the legs of my load-balanced IIS farm, when I go to www.website.com, I get the website.
On one of the legs of my farm, I get redirected to the wfe.website.com address, which requires internet access.
My colleagues do not have the same problem.

For troubleshooting purposes I really need to be able to access this site locally.