@Scott ********
, having a child with a thai woman out of wedlock unfortunately grants you no parental rights as the father. It doesn't matter if your name is on the birth certificate or the child has a passport from your country, you have no parental rights and can't use the child to get a yearly extension of stay based on supporting a half thai child here.
You would have to go thru family court and get granted parental rights, which if the mother agrees to it is not that hard to do but takes about 4 to 6 months to get thru the court system.
You could get a 90 day single entry Non-O visa from the thai consulate in Savannakhet Lao with just your name on the birth certificate, the mother's thai i/d and the child's house book listing. BUT you couldn't get a yearly extension inside the country at the immigration office. You could only get a 60 day visit family extension until you went thru family court.
I know foreigners who have kids by thai women they weren't married to who have been getting those 90 day Non-O visas and the 60 day extensions for years and years.