Retrieve a file from browser using API gateway, AWS LAMBDA and S3
Here is how we did it:
Check and Redirect:
API Gateway --> Lambda (return 302) 
Deliver Content:
CloudFront --> S3
Check for S3 existence with Lambda returning a 302 to cloudfront. You can also return Signed URL from Lambda with a valid time to access the URL from CloudFront.
Hope it helps.
The setup is using S3 as a storage, API gateway for the rest endpoint and Lambda (Python) for get/fetch of file in S3.
I'm using Boto3 for the Lambda function (Python) to check if the file exists in S3, and I was able to download it but being stored in Lambda machine ("/tmp"). The API Gateway can trigger the lambda function already. Is there a way that once the lambda function is triggered then the download will happen in the browser? 
Thanks!