Update:
So my wife went to the Thai embassy in Paris and obtained the two documents which were required for me to renew our 12 year-old son’s passport in Thailand in her absence:
Letter of consent (หนังสือให้ความยินยอม)
Power of attorney (หนังสือมอบอำนาจ)
I would advise calling the embassy beforehand to check exactly which documents are required, in order to avoid any unpleasant surprises. In our case, the Paris embassy needed five sets of documents and a pre-purchased envelope from the Post Office, though these two requirements were not specified on the checklist shown on their website. The envelope is used to send the documents at a later time as they need to be approved by an upper-level manager who isn’t on premises.
My wife received the documents four days after her visit to the embassy. She sent them to me in Thailand via express post with tracking – the price was 65 euros (2,500 baht), which I think was worth it as I received them within 3 working days and could follow the process step by step online. Then off I merrily went to renew my son’s passport at the local government office, which all went very smoothly, the price was 1,000 baht.