Can't connect Claude Code in VS Code to Claude chrome extension

Environment

• OS: Windows (win32)
• Claude Code: v2.1.23
• Chrome Extension: v1.0.63 (fresh install)
• Node.js: Installed at C:\Program Files\nodejs\node.exe

## Native Host Status (all verified present)

• Registry key: ✓ HKCU\...\com.anthropic.claude_code_browser_extension
• Manifest JSON: ✓ %APPDATA%\Claude Code\ChromeNativeHost\
• Batch file: ✓ ~/.claude/chrome/chrome-native-host.bat
• Extension ID: ✓ fcoeoabgfenejglbffodgkkbkcdhcgfn

## Symptoms

• /chrome command shows "connected and enabled by default"
• MCP tools return "Browser extension is not connected"
• Extension service worker console: empty (no errors)
• Reinstall + multiple Chrome restarts did not resolve

## Expected
Extension should connect to native messaging host