I can't help or advise on the visa or needs for the kid. BUT you need to bring with you the ORIGINAL birth certificate of your kid. Without it (ok, I admit it'd about 20 years ago) they would let us pass immigration to exit the country with thr daughter (born in thailand)
no need to know. As Tod said, take both passports plus the letter from embassy to immigration. Its done in 10 min & dont cost you a single baht. Have done it here (in chiang mai at immigration officr at the airport) already (this spring)
With airlines cxl flights still last minute I wonder of its worth the risk. If you hv time and funds at least plan for an extension as this possibility might become necessarry