I think that it's short for Beth dych chi'n ei feddwl "What are you in its thinking" for "What do you think" or "What is it that you are thinking (it)" with ei for the resumptive "it", which causes soft mutation.
Everybody pronounces it like should it's just that since the TTS is set to Welsh it will pronounce everything using Welsh spelling rules and so the "g" is getting pronounced.