When defining a REST API with API gateway, we can't currently define a collection of models that can be used to map response payloads from an integration to what is returned to a client.

https://docs.aws.amazon.com/apigateway/latest/developerguide/models-mappings.html

#1572 introduces the base support for MethodResponses, but only gives the functionality to use the default provided Empty and Error models that are generated, when you create a REST API. This work should be continued to provide full support for defining models.