I used to work for a company that made baggage handling equipment. They have this device euphemistically called a "PUSHER". It had large paddles which knock the bag off of the conveyor - "POW" - it goes flying in the air and hits a rubber curtain and drops onto the conveyor below.

So when you have packed you guitar just throw it out of a second floor window to see if you packed it good enough.

I would use a hard case that is well lined. Don't put anything else in the case, you don't want a capo bouncing around in there. Then pack the case in a heavy cardboard box (You will probably have to make your own) with plenty of bubble wrap or other padding.

If you don't feel confident enough to kick the bag downstairs then you haven't packed it properly.

Bert.

P.S. You could try to make prior arrangements to carry it on the plane. They will sometimes agree.