Put the PC in the trunk, it will be fine. Maybe put some bags/towels on top of it so it can't bounce up if you guys hit a really big bump, but that is about it. Far more robust then they seem.
When you finally get to your destination, open it up and give it a quick inspection before turning it on. You would want to check for loose components or cables.
Back in the day we used to take machines to actual lan parties and events. Indianapolis, Chicago, Louisville and our own local stuff. I've had machines in the backs of pickup trucks, hatchbacks, huge sedans. Haven't lost one yet.
Chicago I did have to reseat the memory, but that was it. Still have that machine.
Now GPUs weren't quite the monsters they are today, but most of the mid-range ones would be similar in weight. As long as you have the motherboard secure and the GPU slotted in well and secure it isn't going to go anywhere.
As USAFRet mentions, it has survived way more punishment in getting to you. If you are REALLY worried, like I said, take the GPU out and put it in its box.
Stock CPU cooler should be fine.