How to troubleshoot that my local web handler correctly returns the payload (status, body,etc,,) but deployed (api-gateway warpped) function doesn't?

what kind of API gateway are you using, REST or HTTP?

Here is an explanation between their difference:
https://docs.aws.amazon.com/apigateway/latest/developerguide/http-api-vs-rest.html

If you are using the newer style HTTP one, you have to explicitly configure the lambda integration, you are assigning to a route, to use the version 1.0 payload format.

That’s a common problem, so I’m mentioning it without looking into the details you shared.

What was the response you got?
I’m offered to download something, when I clicked on your API gateway link, but not sure what, because I just tried it on an iOS device.