Just sending a made-up session id isn't going to do anything useful. As Eric pointed out, the browser will take care of sending the session cookie. Just let it do its thing.
Actually I am doing a single sign in two different domains, after successful logged in, whenever mobile browser communicating with one domain, we have to send the background request (AJAX call) with cookie to other domain to keep the session alive. In case of form submission browser able to manage the cookies for the domain and send in request. However AJAX call fails to send the cookies.