All I can tell you is in Chiang Mai, they did for me.
Brought the paper to immigration and it this was good. Each year now, at extension time, I go get a copy of the letter from the office and get my extension. I NEVER went to court.
If I have picture of the letter, I will send you so you can see.
I am responding to you as someone who actually did this years ago.
All I had to do was go to the Amphur office with a couple witnesses and they provided me with a letter that immigration accepted and have gotten my extension every year with no problem. Everyone said I need to go to court. I did not. Of course my name is on the birth certificate.
I think it was free or very little money. My son was born in Thailand and I'm not married to his mom. If you want more info, PM me.
FYI - in Chiang Mai I just did this. I did NOT need to go to family court. I just needed a letter from the government building in Mae Rim that stayed I was the father. I had to bring 2 witnesses other then the mother with me. Also needed a resident certificate, pics and the usual copies.