is it possible to pass Apache's REMOTE_USER from one Apache server to another?
What do you think would happen if configure my .hpasswd file with a user littleK and password of my choice and then passed the request on to Stackoverflow?
That's just not how the web works. And one of the many reasons that Basic HTTP authentication is a really bad idea.
You need to go read up on authentication in web applications, state management and how HTTP handles cookies.