Expose remote session URL/token in status line stdin JSON

Summary

Claude Code remote/web sessions have a stable URL like https://claude.ai/code/session_01XUTbmXVVNk1zphmMtkfzAE, but this token is not surfaced to status line scripts. The local session_id (a UUID) is unrelated to the remote token, so a status line cannot derive or display the remote URL.

Request

Add a field — e.g. remote_session.url (and/or remote_session.id) — to the JSON piped to the statusLine.command script. Document it at https://code.claude.com/docs/en/statusline alongside the existing session_id, session_name, etc.

Why this matters

• Distinguishing remote sessions at a glance (the local UUID looks identical to a non-remote session's ID).
• Quickly reopening a remote session in the browser.
• Sharing/linking the active session in chat without copying from the URL bar.

Workaround today

The token is currently readable from ~/.claude/sessions/<pid>.json as bridgeSessionId, but this is undocumented internal state and not safe to depend on.

Environment

• Claude Code 2.1.123 (macOS, darwin 25.2.0)