How about putting it straight up and down into a pannier and then using bungies to secure the middle to the rack to keep it from moving around too much or bouncing out?

You could also try going perpendicular to your rack, I have done it with other instruments. But you have to be very, very aware of how wide you are and make sure that you have the weight distribution figured out. I also had panniers on at least one side to help create a little wider platform.

Do you already have a trailer? I think you could make that work as well. You might need to play with the angles a bit and secure it with some tie downs to keep it in a position that works.

Do they make backpack style cases or case covers? I used to carry my viola on my back, I hooked up the strap to get it messenger style. It kind of worked but I sometimes got caught on the seat or smacked in the head. Your case might be too long to try that but you also might be able to get a better angle than I could since your case would not be as wide.