A possible alternative outing is to visit Mitsuwa Market in the Palms/East Santa Monica area. While certainly not Thai, it has more than purely Japanese groceries, and is about a 15 minute drive (9 miles) from LAX in no traffic. It's just west of I-405, near Venice Blvd, and Sawtelle.

http://www.mitsuwa.com/tenpo/sant/eindex.html

As others have said, getting to/from Thai Town in your window is not workable. Also, be aware that taxi fares from LAX are not low. You would probably also need to have the cab wait for you, as it would be impossible to flag one in that area, when you are ready to leave. My guess is with a half hour of waiting time at the Mitsuwa store, you would run $60, before tip.