Steve, interesting that this question has come up now as I had intended to ask the same, but for travelling in the opposite direction, how difficult to fly in and buy a bike in the States etc. Due to the pandemic amongst other things, any trips are on hold for the moment anyway so I'll save the question for another time.
You might want to check out
www.horizonsunlimited.com for useful info on all things concerning world travel on two wheels. The site is run by a couple of experienced world travellers ( Canadiens) and their mission is to compile as much useful info as possible and share it with other bikers/travelers.
Back in 2011 I shipped a bike and trailer from the UK to the USA and it cost £2500/$4000 return. For the six month trip it was worth the expense, for anything less I'm not so sure. It was a lot of hassle, worry and red tape, you are giving your bike over to the shipping company weeks beforehand and delays can be quite common, ships get stuck out at sea etc. Mine was delayed at sea for one week, But then customs didn't visit the port for another week so no bike release, resulting in a two week delay eating into your valuable travel time.
Flying your bike would seem to be less hassle but then you may be required to partly dismantle the bike and arrange for crating it. It would probably cost more to fly it but then you wouldn't have the exorbitant port fees to cover if you had shipped. $320 dollars to roll a 150 pound trailer off a 4 inch high pallet on top of all the other port fees. Had to be paid in cash and six guys turned up to help, go figure.
So, given all this agro, my solution would be to find someone in Europe who has a bike that suits you, find out if they are planning on coming to the States, and do a bike swap. People do it with houses all the time. If you are too precious with your own bike to let others use it, then buy another, what the heck, you were going to buy one in Europe anyway and then you sell it on afterwards, you may lose a little on the resale but I'm sure that would pale in comparison with the cost of all of the other options and it has to be easier to buy and sell a bike in your own country than trying to do from 6000 miles away.
Right.....so when do I book my flights ( just kidding )