I've run it through a simulator, the 302 redirects to bing.com. It's got nothing to do with the header fields, it's my account... puzzling.
I've tried 3 different browsers. I can get the https://www.bing.com/toolbox/webmaster/ page with the Sign-In button. The first step is to enter email address, OK that works and takes me to the next page where I enter my password. It's when I click OK there that I'm redirected to the Bing Homepage instead of my dashboard.