I have a script on my page that is dealing with session timeouts, redirecting the user on the client side when the session is due to expire. The complete code is somewhat more
I was running into the same type of issue, and I created a different question so I could focus in on what I perceived to be the heart of the issue. (IE not repainting while computer is in lock screen.)
Anyway, I finally figured out a way to resolve this. You can see my answer here. Essentially you can fix your issue by putting some JavaScript inside your auto-logout destination page to periodically update the page content.... and this will force IE to repaint the page.
This way I can execute any Auto Saving logic I want right before their session expires on their current page, and then kick them to the Auto-Logout page.