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.
We faced this exact issue. We are using Ansible for our Infrastructure. Could apply to CLI or Cloudformation or even the SDK
The solution to our problem was to make sure that the Lambda policy was defined in a granular manner for the endpoint verbs in API Gateway for the lambda you are attempting to use.
For instance, We had multiple routes. Each route(or sets of routes) needs its own lambda policy defined that allows lambda:InvokeFunction. This is defined in the Lambda Policy module for Ansible. With this, the lambda trigger was enabled automatically.