So it turns out that the session doesn't need to be stored as a cookie and that the authentication will work off the network but just requires browser settings to be changed. I tested these steps out and it worked for me.
-
Firefox:
- Open a new tab
- Type 'about:config', minus the quotes, into the address bar
- Promise firefox that you'll be careful
- Search for 'network.automatic-ntlm-auth.trusted-uris', minus the quotes
- Double click the item
- Add [insert server address here] to the string
-
Chrome & IE:
- Open IE
- Go to Tools > Internet Options > Security Tab > Select Local Intranet > Click the Sites button > Advanced > Add https://[insert server address here] > Close > OK > OK
- Restart your browser
I did not test Safari though, so I am unsure if they follow the same settings as Chrome or if they also need special configuration. Anyways, this didn't answer my original question but did solve my problem.