Can I get a re-entry permit at Phuket Airport for a 1-year extension based on having a half Thai child?

You get it from Immigration/Passport Control Gate 1 - this is also the Thai booth. Sometimes they let you use that entrance (its the door on the left) otherwise you enter through the foreigner exit and explain you want a re entry permit and you hop under the barriers to that one. Make sure you have the fee - normally 300-400 THB more than immigration. And don’t forget the photo! Takes about 15 minutes which is way quicker than the immigration office took me. They have always been super helpful and nice

Post on a Phuket specific group and ask someone will know the answer

That's a great question, and I don't think anyone knows. I recommend getting a re-entry permit at the local immigration office.