Which solution will meet these requirements in the MOST operationally efficient way?

Explanations:

Configuring mock integrations for API Gateway methods allows developers to simulate the backend responses without needing the actual backend implementation. This method is straightforward and efficient, enabling frontend developers to continue their work without waiting for the backend to be completed.

While integrating a Lambda function with API Gateway to return a mocked response is a viable option, it introduces additional complexity and operational overhead. This solution requires deploying a Lambda function, which may not be necessary if the goal is to allow frontend development to proceed quickly.

Adding new API endpoints to the API Gateway stage that return a mocked response is not the most efficient way to facilitate frontend development. This approach could lead to unnecessary complexity, as it requires modifying the API Gateway configuration each time a new endpoint is needed, rather than using mock integrations for quick and simple simulation.

Configuring a proxy resource for API Gateway API methods is not the ideal solution in this scenario. Proxy resources are typically used to forward requests to another backend service, and while it allows for a more flexible API structure, it does not directly facilitate the quick development of mock responses needed for frontend development.