I have a working AWS API Gateway Method Integration that returns a payload, I’d prefer it to return no payload at all as an HTTP 204 No Content response.

The CloudFormation definition of a working version of the method. It just delegates to SQS to send a message.

# The rest of the Cloudformation template is boiler-plate API Gateway and elided.

  RestPost:
    Type: AWS::ApiGateway::Method
    Properties:
      RestApiId: !Ref RestApi
      ResourceId: !Ref RestResource
      AuthorizationType: NONE  # This method is public.
      HttpMethod: POST
      Integration:
        Type: AWS
        Credentials: !GetAtt Role.Arn
        IntegrationHttpMethod: POST
        Uri: !Sub arn:aws:apigateway:${AWS::Region}:sqs:path/${AWS::AccountId}/${Queue.QueueName}
        PassthroughBehavior: NEVER
        RequestParameters:
          integration.request.header.Content-Type: "'application/x-www-form-urlencoded'"
        RequestTemplates:
          application/json: Action=SendMessage&MessageBody=$util.urlEncode($input.body)
        IntegrationResponses:
          - StatusCode: 200  # This, the first one, is also the default match, so the code here means nothing
            ResponseTemplates:
              application/json: |
                {}  ## a minimal JSON document
      MethodResponses:
        - StatusCode: 200  # This, the first one, is also the default match, so the code here means nothing

Enabling X-Ray has indeed shown that there is a fault in the method, but I cannot seem to extract any other useful information from it.

Change the ResponseTemplate to

#set($context.responseOverride.status = 202)
{}  ## a minimal JSON document

This works and correctly maps the status code from 200 OK to 202 Accepted.

Change the ResponseTemplate to

#set($context.responseOverride.status = 204)
{}  ## a minimal JSON document

This returns a 500 Internal Server Error.