To obtain a visa based on children born in Thailand to a Thai mother, a birth certificate is required. If you are still unmarried and want the yearly extension, based on children, you need to obtain paternity rights from your local amphur.
In that case you are right, if they require this in phuket, definitely better do as you say. Each immigration has its own rules and also land of transport. Better to pay 300 baht than to spend flights, hotels etc. in addition to make an appointment and lose even 1 week of time.