Might be better off building a new one in the US and shipping it with your other stuff. Sales tax (import tax?) is pretty bad for electronics in the EU, stuff is a lot more expensive. Everything costs pretty much ~30% more.
You can package the computer in the box of the case and maybe take out the gpu and/or the cpu cooler and pack that separately so it doesn’t break in shipping.
How much stuff are you bringing? Are you getting like a shipping container for furniture, etc. or just essentials? Are you staying in one spot for long? If not, gaming laptops might suit you better (once again, cheaper to buy in the US).
That makes sense then, I wouldn’t trust the plane luggage handlers with the pc. In that case you’re probably right to sell. I would still price compare for a gpu or cpu and get the rest of the build later, but also heavily factor in the hassle of carrying stuff and basically not having a warranty for parts that you bought in the US.
I did actually move with my gaming pc twice. But I had most of my stuff shipped in a truck and only the essentials on the plane. You’ll probably end up having limited space/weight in the checked luggage anyway. Gool luck with the move!