What you need is a backpack with a frame, so you have something solid to attach your bag to....An alice pack or swiss "engineers" pack would be much better. Both can be had for under 50 bucks.
As far as attachment, compression straps are the way to go. You can get them much tighter than bungie cords. As far as attaching the sleeping bag to your pack--if you attach it at the bottom, it's probably going to flop around your butt and be very awkward/ unstable. The pack would probably carry better if you attached the bag to the top or had it laying vertically, along the middle of the pack. Also, you can always sew on attachment loops anywhere you want to lash things.
The military sleep system is excellent, though. Your bedroll is one of your most important pieces of gear, so get the best that you can afford.. Having traveled without a blanket or a sleeping bag, i can tell you that it SUCKS. A bag that is warm down to freezing (at least), big enough for you to crawl into, and will dry fast is a good bag.