When I used my bike to go to work and back and I had a large computer briefcase, I would just strap it to the passenger seat using three bungee cords. Worked like a charm. That way I wouldn't have to buy a specific case for the bike and when I got to my destination, I could take the three cords off an have a regular computer case to carry around instead of special bike bag.
Anyways, so my suggestion is just try strapping your current backpack to the passenger seat with bungee cords. You may have play around with them for a while to find which bungee locations work best for your backpack.