[FEATURE] Support /status in VSCode panel mode

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

I use the Claude Code extension in VSCode and I want to verify the config with /status like in the CLI.
For example for me it is important which API provider/backend is used and configured by the extension.

Proposed Solution

Support the /status command or another hint about which backend/api provider is configured and used by the extension.

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Configuration and settings

Use Case Example

_No response_

Additional Context

_No response_