I would discourage you from shipping the GPU, HDD, and SSD wrapped in bubble wrap and stored inside the PC case inside the original PC case box. You would really need to pack the space full with bubble wrap so that nothing can move. Still, in shipping things will settle. The general rule for shipping things to see if it's OK or not is to package it up then drop it from 3 feet and see what happens. I'm not sure I'd want to try that, so I would ship the GPU and HDD separately in a lot of bubble wrap and a tight fitting box. If something breaks by shipping everything together it's going to cost more than the cost to ship a second package, plus you're paying for the extra weight even when you combine it so the savings may not be all that much compared to the risk.
Most importantly, you should remove the heatsink and wrap that up too. The SSD I actually don't think needs to be wrapped, you could just tape the toolless caddy to the inside of the case and it shouldn't go anywhere. It's solid state so you don't have to worry about vibration.
Also make sure the wiring is good and secure with some zip ties if you didn't already do that. Loose cables can cause scratches in shipping.
Depending on which country you're shipping to, you may also want to encrypt your storage first.