feat: sync device timezone to backend on startup

On shell mount, reads the IANA timezone from the device via
flutter_timezone and POSTs it to PUT /api/settings as user_timezone.
Mirrors the web app's App.vue behaviour so briefing events and the
chat route use the correct local time on Android.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

pubspec.yaml

@@ -26,6 +26,7 @@ dependencies:
   flutter_inappwebview: ^6.1.5
   flutter_markdown_plus: ^1.0.7
   google_fonts: ^8.0.2
+  flutter_timezone: ^5.0.2
 
 dev_dependencies:
   flutter_test: