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.