As the link says, registering a birth in Thailand is unnecessary. A translated copy of the Thai birth certificate is fine when applying for the child's UK passport.
says. Not necessary to register the birth. A translation of the Thai birth certificate will be needed eventually when you apply for the child's UK passport.
It helps if it's a UK person who countersigns, but it is not essential. The Thai doctor who dealt with the birth, or your current pediatrician, might be worth trying. Edit: it seems doctors are specifically excluded from being counter-signatories, so my suggestion probably won't work.
That is NOT correct for UK passports. There is no luck involved. Either you (as the British parent) are eligible to pass on your citizenship, or you're not. As stated earlier, the Thai MFA do not have to legalise the Thai birth certificate. It is checked when you submit the application and documents at the HMPO office at VFS in Bangkok.
You just apply for her UK passport via VFS in Sukhumvit soi 13. You will need a translation of her Thai birth certificate (showing your name) but it does not have to be stamped by the Thai MFA. Most importantly you need YOUR long-form birth certificate showing that you are British Otherwise Than By Descent. You will also need the mother's passport/birth certificate/ID and, if you're married, your wedding certificate (translated if in Thai).