It didn't happen to myself yet, but to other members of the French FB group. It seems that the family doctor has access to our files with the social security and can print out the two detailed certificates.
Immigration doesn't extend visas. Immigration extend the time you were allowed to stay upon arrival. It's called "extension of stay" and works perfectly with visa exemption entries.
I am in the same situation. My worry is if immigration will accept the seasoning money as I understood that money should be coming from abroad for the first extension.
In other words, you want an extension to stay in Thailand with a wife... who will not be in Thailand. That's how immigration will understand your request.