your information is incorrect. I am currently on a Ed Visa and was able to obtain a Thai bank account with a debit card. it wasn't easy as the first two Banks said no only University degree qualifies but there are banks out there that will give you just have to do the work and find them.
The statement therefore is false. my guess would be to start with immigration. show them your marriage license or whatever you have that shows that you are married. if that doesn't work then I would suggest going to an agent. sounds like they just want to be sure that the child is yours and that you were properly married.
I'll preface it by saying I don't know for sure. However, I was doing a lot of research on it yesterday and today and it depends on what country you're from. Some countries need a background police check but many do not. I was going to do an Non-Immigrant O Visa but now I'm thinking about doing a education visa and I'm 66. I found a school down near Asoke. They handle all the paperwork to get the agreement with the ministry of education and then will give that to you and you go to immigration with the required photos and other paperwork. So I'll pass what I know on but I'm sure somebody will correct me.