You can definitely do it, but you will need to buy a separate seat for the fish.

Actually, if the fish is not too big, you should be able to just put it in a bag with water, make sure it is aerated, and but it in your hand carry. Most tropical fish get sent that way and the fish can survive for at least several days.

Importation or quarantine could be a bigger issue. You might also want to talk to a tropical fish exporter. There are many in Thailand. They can probably send it fedex and know how to deal with U.S. importation issues.