Using a bike box for any trip should be a good call if you pack it properly. Most airlines charge now, check how much. They also state a size and weight limit. Less than 23kg and dimensions are a standard and as long as you stay under it you will be ok.
Going to the US it's normally classed as a 2nd piece of baggage, so united airways is about $100 each way, BA is ??34 each way.
If you are in Europe then the low cost carriers will charge a fixed fee, easy jet is about ??30 each way but also state that the bike will be left if there is too much baggage !! Not seen it happen but bikes go awry so make sure it's booked and well signed.
As for shipping, never used it as its generally a rce organiser thing and most of the biggest races in the US offer it. However if you can get a carrier to ship your bike to a local bike shop near the race venue and then build it for you and repack on the return for a good price I would go for it.
The other consideration is who you are travelling with, a bag of kit, bike box and hand luggage can be a drag if you want ease of travel.
The choice is yours,
