There are specialized "transport bags" for bikes. The shape is trapezoid, not very different from the sidescreen bag, but bigger of course. This might fit onto your luggage rack and both the bike and the car will be protected. It involves a little procedure, as you have to take off the wheels, which goes into the bag one of each side of the frame. You detach the handlebar and let it hang by the brake/gear cables parallel to the frame. Finally you take off the pedals so nothing is sticking out. Sounds a hassle, but it doesn't take many minutes provided you have the tools at hand and some training. Living in a bike country such a bag should be easy to find.
