Alternative : When you arrive, you can go straight to domestic departure and then book your flight to CM. If no flight available then take a taxi to Pattaya.
There are several companies desserving CM from BKK : One2Go, Airasia, Orient, bangkokair.... I don't remember all of them, you must check them all.
A taxi ride to Pattaya is about 2,5 hours from the airport.

Have a nice trip.