I wonder if hidden iframe pointing to (the start page of?) the logon server would work. Then the browser would send request to the server (asking for the iframe's content), and the server would send a cookie along with the document. And then Javascript would submit the form.

That's in theory. I've personally never done that before. If you'll get this working, be so kind and let us know! I for one am really curious. BTW, it does look like a good way to steal other peoples passwords :)