I don't use any additional padding when I pack my bike. I leave the axles in the hubs. I put the frame in upside-down, drop the fork, crankarms, and pedals in through the rear triangle, and squeeze the drops down the side firmly against the (inflated) tires. It is a very snug fit, but everything fits. I should've snapped a photo before I unpacked it the other day. I also throw my tools in there.

It should only take you a minute or two to get your headset adjusted properly when reassembling. And, like hyperRevue said, you don't need to remove the headset itself, whether threaded or headset (I've travelled with both).