[Feature] Register Claude Code as VSCode Language Model provider to enable Agents window integration

Problem

Users with a Claude subscription cannot use the VSCode Agents window because it requires a GitHub Copilot subscription. The Agents window uses the VSCode Language Model API (\scode.lm\), which currently has no Claude provider registered by the Claude Code extension.

When selecting a model in the Agents window, only "Auto" appears — and sending a prompt results in "Language model unavailable".

Expected behavior

Claude Code should register itself as a \scode.lm\ language model provider, so that users with a Claude subscription can use the Agents window natively — without needing GitHub Copilot.

References

• VSCode Agents window documentation
• VSCode Language Model Chat Provider API

Prior art

• A community extension (claude-model-provider) already demonstrates this is possible by registering Claude models via the \scode.lm\ API. However, its setup instructions reference the 'Chat: Manage Language Models' command which no longer exists in recent VSCode versions, and it is unclear whether it works with the Agents window specifically. The repo also has issues and discussions disabled, despite the README inviting users to file issues there.
• Related feature request: #66521 (registering as inline chat edit tool provider)

Notes

The VSCode \scode.lm\ API is already open to third-party providers — this is purely an implementation task for the Claude Code extension.