Add Terraform setup for Lambda using Mangum and API Gateway

[Subhransu-De]

What to build

Add a Terraform infrastructure setup under /infra that deploys this FastAPI template to AWS Lambda using Mangum behind API Gateway HTTP API. This setup represents the common ASGI-adapter approach for running FastAPI on Lambda.

This issue may include the minimal app-side Lambda handler needed for Mangum, while keeping the normal local/Docker runtime intact.

Acceptance criteria

• Terraform provisions Lambda, IAM role, API Gateway HTTP API, Lambda integration, permissions, and CloudWatch logging.
• The FastAPI app can be invoked through API Gateway using a catch-all route.
• Mangum integration is added without breaking the existing Docker and local development entrypoints.
• Lambda timeout, memory, architecture, environment variables, and secret references are configurable.
• Documentation explains how this differs from the Lambda Web Adapter setup.
• Documentation calls out Lambda-specific caveats around FastAPI lifespan events, app startup migrations, cold starts, and DB pooling.
• Outputs include API endpoint URL, Lambda function name, API Gateway ID, and log group name.
• A verification command is documented for checking the deployed health endpoint.

Blocked by

None - can start immediately.