I'm in the US military and stationed in Europe. I'm quite experienced with this...when I travel for work longer than 4 days (either here in Europe or back to the US), I pack a bike into a bag and fly it as baggage. I've done it successfully more than 15 times over the last 4 years. Some airlines charge, many do not (American and Delta do not, Lufthansa routinely won't, Air France is hit or miss, but mostly miss). Know the rules before you get to the airport....and in this COVID time, you might be able to get them to waive fees since you're flying and most people are not. I have, I'm sure you could too, just be reasonable as you discuss it, and ask to speak to a supervisor if necessary. Also, some airlines require you to notify in advance if you are bringing a large item, or they may refuse it at the counter (KLM, sometimes).
First thing. DO NOT SHIP IT! It costs more than flying with it over the ocean if you use a freight service like FedEx, UPS, DHL, etc. USPS is cheaper, but not by much, and they are very slow, 4-6 weeks based on people I know who have done it, pre-COVID, I can't imagine they've gotten faster.
Fly with it, like I said above. That means you'll drop it at the airport, and when you recover your luggage at the other end, it'll be there. Packed properly, there will be no damage (I'll explain how in a minute). The key is to use a good quality bag. EVOC makes one that requires minimal disassembly,—fork remains in the frame—but is large enough that if the airline charges a bike fee you'll likely have to pay. Orucase makes one that requires more disassembly, but frequently can be slid past the counter with no fees (I've done it without fail when I used one before the bike fees started falling off about a year ago and switched to the EVOC bag). Some people use the box method, I don't personally like it; boxes were designed for transporting semi-built bikes in a cubes configuration where individual boxes are not generally being handled, but whole pallets of them. To fly with a box, you have to ensure it is padded and buffered well, or it will get jostled enough to cause damage, cosmetic or worse. I've seen it on other peoples' bikes, and it's why I only use bags. The other reason to use a bag is that TSA will absolutely open the piece, and they may not resecure the box, but a bag with zippers will get zipped back up...this TSA bit is why you need to pay attention to the info about padding below.
Disassembly. If you are not sure exactly how, buy your favorite mechanic a beer and have them talk/walk you through the process. And, then allow yourself a good hour to do it. General stops: (1) remove pedals, (2) Remove wheels, (3) If necessary for the bag, remove chain and rear derailleur, (4) pad the frame, (5) remove the handlebars, pad the stem, or drop the fork, and pad the fork legs, (6) pack it into the bag (YouTube can help on this as well).
Depending on the bag type you use, you may have to remove the fork, or just the handlebars per #5. When wrapping the frame and fork, I recommend using bubble wrap or styrofoam pipe insulation (pool noodles with a longitudinal slit work well). Apply the protection to every member of the frame where it could contact something else, like a fork or handlebar that has been removed and packed into the bag. Pad if you have any doubt, if you don't you'll have chipped paint. Also, if you have disc brakes, REMOVE the rotors, and pack them into a sleeve made from a repurposed cardboard box, ensuring that they do not touch, and are fully encased between layers of cardboard, then taped shut to keep them in the "sleeve." You should NOT need to remove brake calipers or hosing, but as you pack the bike ensure that you do not "kink" any of the hosing near the handlebars. The last thing you'll load are the wheels, follow the bag manufacturer directions for this...even if they do not state to remove the rotors, remove them, or they'll be bent. It sounds a lot more complex than it really is, and I've done it so many times I often am doing the packing of the bike last thing at 8pm the night before a 6am flight. Reassembly is much faster than disassembly, as well.
Happy to answer any questions.