I\'ve been trying to work out how to express (in cloudformation) an API Gateway Resource that has a Lambda function integration type using the Lambda Proxy integration.
I have solved this same issue by simple changing the
Integration:
Type: AWS_PROXY
To
Integration:
Type: AWS
The cloud formation documentation currently is scarce and the API gateway cloudformation documentation doesn't match up to what can be seen on the console which hinders anyone who is trying to resolve an issue.
Hope this helps!