Adding custom response Headers to APIException

Try overriding finalize_response in your rest framework view:

def finalize_response(self, request, *args, **kwargs):
    response = super(SomeAPIView, self).finalize_response(request, *args, **kwargs)
    response['WWW-Authenticate'] = 'Token'
    return response

Edit:

After seeing your update, I think your override of handle_exception should work, I would only add an else statement to call the parent method to cover other exceptions. One thing I noticed in overriding dispatch, which may not be an issue here, is that setting a new key/value for self.headers resulted in a server error that I didn't take the time to track down. Anyways, it seems you are on the right track.