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/providers/api_client_provider.dart

@@ -10,6 +10,7 @@ import '../data/api/milestones_api.dart';
 import '../data/api/notes_api.dart';
 import '../data/api/projects_api.dart';
 import '../data/api/quick_capture_api.dart';
+import '../data/api/settings_api.dart';
 import '../data/api/tasks_api.dart';
 import '../data/repositories/auth_repository.dart';
 import '../data/repositories/chat_repository.dart';
@@ -85,3 +86,7 @@ final milestonesRepositoryProvider = Provider<MilestonesRepository>((ref) {
 final briefingApiProvider = Provider<BriefingApi>((ref) {
   return BriefingApi(ref.watch(dioProvider));
 });
+
+final settingsApiProvider = Provider<SettingsApi>((ref) {
+  return SettingsApi(ref.watch(dioProvider));
+});