Yeah, that would be difficult. I returned to the UK and did it all from there, but we were planning to be there anyway. Maybe getting a UK based friend or relative to do all the posting would work. Either way, it's a pain that it all has to be done in the UK.
I can't speak for New Zealand, but we had to get our UK marriage certificate legalised and then certified in the UK before we could use it for getting dependent extensions. Hopefully your Embassy can do it here!
If you get the stamp transferred and the new Extension at the same time, it adds at least an hour to the time you have to spend there, as you have to get a queue number for the transfer first (N2) and then a queue number for the Extension (N1). We got both queue numbers at the same time, but the Extension numer was called before we'd finished doing the transfer, and was a little chaotic.
If you don't plan leaving Thailand for a while, the child doesn't need any stamp in their passport. The first time they leave you will need to show the Thai birth certificate at Passport Control at the airport and they stamp the passport.
Get the child a passport as soon as you can. They won't need a visa until or if they enter Thailand after leaving for the first time. As others have said, they can enter as visa exempt and then overstay as there is no fine or ban for kids overstaying. However they do get an overstay stamp in their passport.