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>

lib/data/api/settings_api.dart

@@ -0,0 +1,13 @@
+import 'package:dio/dio.dart';
+
+class SettingsApi {
+  const SettingsApi(this._dio);
+  final Dio _dio;
+
+  Future<void> syncTimezone(String ianaTimezone) async {
+    await _dio.put<void>(
+      '/api/settings',
+      data: {'user_timezone': ianaTimezone},
+    );
+  }
+}