How to carry a large backpack on a bike rack?
I too had this problem and found a somewhat unique solution. I explored using a pannier, but since I carry a laptop daily, I was worried about what would happen if the bike slipped on the ice and crushed my laptop bag (something that actually happened to me as a teenager) 

Like you stated, baskets are too tall and cumbersome, so instead I found a 19 by 13 inch metal serving tray and mounted it with heavy duty rubber straps. It even matches the color of my bike panniers so I think it's a good solution. It's very freeing to be able to carry so much on my bike without the constant weight on my shoulders.

 

Since the laptop sits at the bottom of the backpack, I feel like the metal walls of the tray should absorb the shock of a slip or fall and protect it inside. So far I've received multiple compliments on this design. It's been over a year now on my bike and has held up well.