Yeah, there's only one flight a day and not every day of the week. I use it a lot as my wife's family live in Pakse. I hate long bus journeys, so if I ever had to get a visa from Savannakhet I think I'd use it.
You can fly to Savannakhet from BKK on Lao Airlines. It used to be a direct flight (about 1 hour) but now it goes via Pakse, so takes a bit longer. Not cheap, but quick and easy.
She is supposed to go in person. I went once to get permits for my wife and kid, and got referred to a manager with whom I pleaded. She was stern, but relented and I got the permits. Best that she goes if you want to avoid hassle, or get it at the airport.
Then I'd wait until you come to Thailand, then get his Thai and UK passports while you're here. Both are quite straightforward to do here. You'll need YOUR long-form birth certificate to get him a UK passport, so bring it with you to Thailand. Good luck.
Where are you now? He'll need a Thai passport to enter Thailand (if you're currently in the UK) but if you're in Thailand, he'll need a UK visa to enter the U.K. if he only has a Thai passport. I'd get both passports, wherever you are.
Once the baby is registered you will still need to apply for a passport which requires all the same documents. There is no requirement to register births overseas; getting a UK passport establishes citizenship.