[FEATURE] VS Code Extension: Local API proxy for Remote-SSH like Copilot

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

As an ML researcher using a remote cluster, I am having problems with cluster firewall blocks api.anthropic.com:443 (ECONNRESET)—Claude Code fails. GitHub Copilot works perfectly: runs locally, agents edit/execute remotely.

This is critical due to the great computational requirements that it needs that are only supplied in the cluster.

<img width="428" height="167" alt="Image" src="https://github.com/user-attachments/assets/41161f22-5e7f-49d7-b2b3-527029957d7d" />

<img width="934" height="418" alt="Image" src="https://github.com/user-attachments/assets/200d4a93-4aab-4cec-9b15-103d5b06a180" />

Proposed Solution

I'd like to run VS Code extension proxy API calls locally (Copilot-style), while executing the changes in remote.

Alternative Solutions

As alternatives I've tried

• sshfs + local Claude: No live exec (local CPU explodes on ML training)
• Tinyproxy/SSH tunnel: Flaky (400/CONNECT fails)
• Admin whitelist: Slow/policy issues

Currently what has worked for me has been sshfs + local Claude, but I can't test the code since it heavily rely on computational and GPU resources that I do not have in my machine.

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

Example scenario:

1. Install Claude Code CLI/extension locally + cluster (Remote-SSH).
2. Try Claude Code in the cluster (If it works with no blocking, the personal workflow continues from here).
3. If it fails in the cluster, I modify the JSON config to allow Claude Code to run on the local machine but having access to the remote files.
4. Claude Code can make all the changes just as if it was installed originally in the cluster, being able to run agent, accessing files and doing inline edits.

Additional Context

_No response_