As you said, the second and third national diplomas are not recognized.
Civil servants are definitely not allowed to take the exam.
China does not recognize the Singapore private school diploma, but it is recognized internationally.
I recommend a website "People's Republic of China (PRC) Foreign Information Supervision Network" to you. Go there and have a look. There is a list of schools that can be certified to study abroad, not only from many countries in Singapore. I wonder if you are looking for an intermediary?
It is best not to believe all the words of the intermediary. I've suffered a lot about whether my academic qualifications can be certified, so I must be more careful.
The intermediary has always said that the academic qualifications will be recognized immediately, but this sentence has not moved for a long time.