[Feature Request] Support Local Development and Debugging for WebSocket Lambda Streaming service

## Describe the solution you'd like

A streamlined method for locally developing and debugging WebSocket-based AWS Lambda functions that support real-time streaming with the frontend chat. The solution should allow developers to simulate WebSocket connections, send/receive messages, and observe function behavior without deploying to AWS.

## Why the solution needed

Current development workflow for WebSocket Lambda functions requires frequent deployments, making iteration cycles slow.
Debugging WebSocket interactions in a local environment can help catch issues earlier and improve productivity.
Enables seamless testing of WebSocket messaging before deploying to AWS.

## Additional context

AWS SAM CLI provides some support, but WebSocket event testing remains cumbersome.

## Implementation feasibility

Are you willing to collaborate with us to discuss the solution, decide on the approach, and assist with the implementation?

- [ X ] Yes, I am able to implement the feature and create a pull request.

No files, tests, or entry points are named. Start by reviewing the repository's Lambda and frontend WebSocket flow, then compare it with AWS SAM CLI's local event support. Done means agreeing on a local development and debugging approach that can simulate connections, send and receive messages, and observe function behavior before deployment.

Written by the indexing model from the issue text.