[FEATURE] Support VSCode's new background agents feature

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

VSCode just announced a new feature for delegating background agents that complete tasks in their own Git worktrees, allowing developers to switch seamlessly between different features and delegate grunt work as needed. It would be great to see the Claude Code extension more deeply integrate with these VSCode features the way Codex has.

Proposed Solution

I think this would require a significant refactor of the current extension to more closely match VSCode's expected chat integration (the way Codex has).

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Developer tools/SDK

Use Case Example

1. I am working on one feature with Claude Code in Plan Mode via the Chat pane.
2. I am satisfied with the implementation plan and delegate the development work to a background agent, which opens a new worktree and begins working in the background.
3. I begin planning a new feature with Claude Code in Plan Mode in the Chat pane.
4. I delegate this second feature to a new background agent, which creates a separate worktree and begins work.
5. I begin working on a third feature myself, unblocked by the other two worktrees. Occasionally, I switch to the background agents' sessions to review their work and course correct.

Additional Context

_No response_