Many years ago, I saw a young girl on an airplane who just flew for her trip with a rabbit on her lap. I think today, if the animal will fit in a carrier case that will fit under the seat in front, you can take a cabin pet.
When I was interested in buying a Flemish rabbit, I noticed that most Flemish breeders would not ship rabbits. Seems to me that shipping live animals, even babies, should be safe.
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A story that's not too much relevant for the OP, but maybe of some interest. In November of 1991, I had to leave Australia. I wanted to take my cat Annie, back to the US. Problem was I didn't know where I'd be living when I left Australia. I left Annie at a animal shipping service that specialized in shipping pet animals overseas. Actually most of the pets they shipped were to Europe/England. They had a kennel in downtown Adelaide, South Australia and another spacious kennel outside of town for animals that had to wait before being shipped. In March of 1992, I had an address to ship Annie to. They then had a week training period..Annie would be placed in a carrier case for longer and longer periods of time to get her used to being confined for a long time in a carrier case. For shipment, Annie went from Adelaide, to Sydney Australia, where she was transferred to a Quantas flight to Hawaii. In Hawaii, she was met by a US custom veterinarian for a health check. This was organized by the Adelaide, South Australia animal shipping service. After approval, she was loaded on another Quantas flight to Los Angeles. I went to US customs and stood in the "Perishables" line to get a customs release, and then to Quantas cargo to get my cat in her carrier cage. I wondered if she was still alive. When I picked up her carrier, I heard a big angry "yowl"...knew I had a live cat! Took her back to my motel room, she wolfed down some food and water, paced around the motel room to check it out, jumped into our bed, burrowed under the covers and slept for about 10 hours. Her airplane trip, including the vet inspection in Hawaii, cost $800. This was 1992 dollars. She was about 12 years old at the time. She lived to be 19 years old. She was sold as a kitten in Whyalla, South Australia for $2.