I wish I could do that with the UK NHS app QR codes and have the confidence they could be read by Thailand Pass. When I scan them they just say 'no usable data found'. Probably why I've heard nothing from Thailand pass since applying a couple of days ago.
Might be worth registering with the myGP app to access your NHS records. It has a new feature called myGP TICKet which will display your 2 dose vaccination status if your GP practice has enabled this. It will also have your full name, dob, NHS number etc. This might be more official looking to present at Bangkok rather than a small hand written card