man... the sg embassy is strange! Sorry I don't know what letter from Singapore means. I assume it might be a spelling mistake and they meant to write letter from Singaporean? I know for dependants under 20 (obviously children are unemployed), the main applicant (parent) write a sponsorship letter stating how they're related and that they will be financially responsible for x in regards to food, accomodation and any other living expenses in Thailand. Print it out, sign with blue pen and rescan. Use ChatGPT for a template.
I don't understand why they would ask you this as you're the main applicant with the bank statements, but who knows- they're odd sometimes
I've tried applying in Singapore with the medical route and it was very difficult. The embassy wanted the same documents you mentioned and Samitivej hospital wouldn't write me a long term plan letter AND provide me with their business licence. Embassy refused to approve without when I emailed them (understandably so). In the end I just gave up and went with soft power (cooking)