her problems apparently resulted from her using an original birth certificate of her son on a Thai authority (be it an embassy or an Immigration office) These Thai authorities only accept foreign documents if they are certified (legalized) by a Thai entity. . . you are fine if you hold an original Thai birth certificate that lists you as being the mother of a "half Thai child" . . you can use it for an application to a "dependent" Non-O visa for your child
her problems apparently resulted from her using an original birth certificate of her son on a Thai authority (be it an embassy or an Immigration office) These Thai authorities only accept foreign documents if they are certified (legalized) by a Thai entity
then why do you tell us an inflated overpriced service? The agent I know in Pattaya will get you BOTH Thai drivers licenses (car & motorcycle) on any visa (even on a visa-exempt!) for 7000 THB, and a blood test is not required, I have no idea who told the OP
the requirements on how to "change" from the tourist visa or from a visa-exempt entry, to the initial "90-days Non-Imm-O Family Dependent Visa", are described in the .pdf under number 8 of the Immigration website regarding the "change of visa type" . . . . print it and take it to Immigration, the requirements are listed both in Thai and English and if you are lucky, Immigration is willing to accept your son doing the "change of visa type" . .
while your 90-days Non-Imm-O Family Visa is a done & dusted things, because you are married to a Thai husband, you son has no family connection to your husband. That's why he only can apply for a 90-days Non-Imm-O "dependent" Visa based on being the son of somebody on a Non-Imm-O Family Visa. Normally he could do the "change of visa type" from a tourist visa to the 90-days Non-Imm-O Dependent Visa on Immigration inside Thailand. However we are aware that some Immigrations won't touch the "change of visa type" process not even with a pair of tweezers. The Immigration who turned you down are one of these Immigrations because they are unsure which are the requirements. . . these are listed under Number 8 on this Immigration website .. . . . But please be aware that this Non-O Visa is limited to children only until the age of 20 . . . . . .
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Your problem is how to change your location and find an Immigration who is willing to accept it
as your British embassy does not certify your income, there is no way around the requirement to transfer a minimum of 400,000 THB into your Thai bank account as soon as you have entered Thailand on a 90-days Non-Imm-O Family Visa, and the money needs to have been sitting in the account for a minimum of 2 months on the day you apply for the "1-year stay permit extension"
the agent I know in Pattaya offers this service for BOTH licenses for 7000 Baht . . . I am very sorry for you if you paid an inflated price of 20,000 Baht
. . . sure, he could do it by himself . . . . I do it by myself since 24 years, but this is advice for a complete newbie, he'd be better off with the help of an agent, it's only 5000-7000 Baht