https://aws.amazon.com/sdk-for-browser/
I'm surprised I haven't seen more discussion of SDK for Javascript since it removes a lot of the server requirements. Have you considered using AWS SDK for Javascript, or do you explicitly want a REST API as an abstraction layer? The API certainly provides some advantages if you ever want to migrate off AWS. In my case, the lambda functions get invoked by particular events on RDS, S3, or manual/browser triggers instead of AWS API Gateway invocation.