task: Migrate all API stuff to AWS Lambdas

See title.

No files, tests, or entry points are named. Start by scanning the repository for the current API layer (route handlers and app startup code) and any deployment or infrastructure config files to map how requests are currently handled. Then identify which endpoints or handlers should move to AWS Lambda and what routing/proxy changes are needed. Done means a clear migration scope and plan for moving API functionality to Lambda, since acceptance criteria are not defined in the issue.

由索引模型根据 Issue 内容生成。