There is no way to do this provided in the http specification. But being ever so stubborn, I've agonized more than once on how to get around this, and even tried some rather bizarre ideas that did not work out. Looking at the site you referenced in your update, I think this bullet point is the key:

• Effectively eliminates the login popup box; works with recent Internet Explorer updates.

This implies that something is different about newer versions of MSIE that provides a way to bypass the standard browser login method. They do not specifically mention any other browsers that this will work with. Anyway, I'm still curious, so if you figure out what they're doing please post an update.