Turkish Airlines, Lufthansa, KLM, Korean Air allow pets in cabin.
I have flown my pugs in cabin (under the seat) on Turkish Airlines and Lufthansa.
You must get an export permit via a USDA accredited veterinarian in the USA and an import permit into Thailand. I have used Relo4Paws for the Thailand permits. Relo4Paws can be found on Facebook.
Google USDA certified vet for where you are in the USA
There are more Airlines that allow Pet in Cabin. Portuguese TAP. Ethiopian. Austrian airlines.
I listed the most convenient between the USA and Thailand
You can also split the trip by using TAP and then Lufthansa, to keep costs down.
I flew pugs to Lisbon on TAP, then flew Paris to BKK on Turkish Airlines
Marc, I will shortly be in the same position. I am American but living in Germany. What did you need to apply as a non-German citizen at the Thai Embassy in Germany E-Visa.