You will want to either get a small styrofoam cooler, a tubberware container that holds at least half a gallon of water, or rinse out a bottle with a large mouth (like a gatorade bottle). Either way, make sure to rinse out the container very very very well. Don't use soap. If you use a gatorade bottle or similar, make sure that you rinse until you can't smell what was in the bottle. Then rinse some more. Use boiling water if you can. Dry completely and then rinse it out some more the day before you leave.
Don't feed your betta for at least 12 hours (up to 36 hours) before the flight. Do a water change the night before. When you are ready to put the fish in its transport, put fresh clean water in the container. Make sure to add your water conditioner. Read the directions for how much to use for a 10 gallon, and subtract until you have roughly the right amount for the size of your container. A little extra conditioner won't hurt, if it isn't Prime.
Catch your betta and place it in the container. Close the lid/cap. Make sure that you bring a small towel or cloth to cover the container with. The betta will be glad for the darkness and less stressed. As soon as you can, acclimate him once you reach your destination. As soon as you can, put him into a regular tank.
Also, you will want to open the container at least twice during the flight to let fresh air in (and also to check on the betta). He should do just fine.
Good luck!
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