This is my (not Peter's) advice for Canadians/Mexicans for TN, Australians for E-3, and seems to apply for Singaporeans/Chileans for H-1B1: If it is an automated system, just put no. That may be an automatic filter. There will be time to talk to a human and say "all we need is X/Y/Z" where X/Y/Z is something the hiring manager can do without involving a lawyer. If a human is asking you, then just say "Yes. I need a visa but I can walk you through the process. No lawyer needed."

hey fellow singaporean - you may enjoy my h1b1 guide https://www.swyx.io/h1b1

most impt thing to note is that while you do still need a visa sponsor, it is a LOT easier and cheaper for you and yes ive had a job where i paid for my own visa lawyer, and only needed simple docs from the company (which was ~10 people at the time if that helps). didnt turn out to be a GREAT job ofc but thats separate.

dont ask me how to get a job in the us from singapore tho. if ur applying thru the "front door" as u seem to be doing, yes its harder. lots more back door options but you'll have to network well to do it (i recommend https://swyx.io/LIP)

Not Peter Roberts nor a lawyer, but: Why don't you just say "I'm Singaporean, so I'm not sure I would need to apply", and give a link to some article which describes the special visa eligibility for Singaporeans? Also note that H-1B1 is just one kind of visa, and there are many others - for which you likely would need to apply.