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.lock

@@ -169,6 +169,14 @@ packages:
       url: "https://pub.dev"
     source: hosted
     version: "2.1.2"
+  equatable:
+    dependency: transitive
+    description:
+      name: equatable
+      sha256: "3e0141505477fd8ad55d6eb4e7776d3fe8430be8e497ccb1521370c3f21a3e2b"
+      url: "https://pub.dev"
+    source: hosted
+    version: "2.0.8"
   fake_async:
     dependency: transitive
     description:
@@ -299,6 +307,14 @@ packages:
     description: flutter
     source: sdk
     version: "0.0.0"
+  flutter_timezone:
+    dependency: "direct main"
+    description:
+      name: flutter_timezone
+      sha256: e8d63f50f2806a3a71a08697286a0369e1d8f0902961327810459871c0bb01c2
+      url: "https://pub.dev"
+    source: hosted
+    version: "5.0.2"
   flutter_web_plugins:
     dependency: transitive
     description: flutter
@@ -640,6 +656,54 @@ packages:
       url: "https://pub.dev"
     source: hosted
     version: "2.3.0"
+  permission_handler:
+    dependency: "direct main"
+    description:
+      name: permission_handler
+      sha256: "59adad729136f01ea9e35a48f5d1395e25cba6cea552249ddbe9cf950f5d7849"
+      url: "https://pub.dev"
+    source: hosted
+    version: "11.4.0"
+  permission_handler_android:
+    dependency: transitive
+    description:
+      name: permission_handler_android
+      sha256: d3971dcdd76182a0c198c096b5db2f0884b0d4196723d21a866fc4cdea057ebc
+      url: "https://pub.dev"
+    source: hosted
+    version: "12.1.0"
+  permission_handler_apple:
+    dependency: transitive
+    description:
+      name: permission_handler_apple
+      sha256: f000131e755c54cf4d84a5d8bd6e4149e262cc31c5a8b1d698de1ac85fa41023
+      url: "https://pub.dev"
+    source: hosted
+    version: "9.4.7"
+  permission_handler_html:
+    dependency: transitive
+    description:
+      name: permission_handler_html
+      sha256: "38f000e83355abb3392140f6bc3030660cfaef189e1f87824facb76300b4ff24"
+      url: "https://pub.dev"
+    source: hosted
+    version: "0.1.3+5"
+  permission_handler_platform_interface:
+    dependency: transitive
+    description:
+      name: permission_handler_platform_interface
+      sha256: eb99b295153abce5d683cac8c02e22faab63e50679b937fa1bf67d58bb282878
+      url: "https://pub.dev"
+    source: hosted
+    version: "4.3.0"
+  permission_handler_windows:
+    dependency: transitive
+    description:
+      name: permission_handler_windows
+      sha256: "1a790728016f79a41216d88672dbc5df30e686e811ad4e698bfc51f76ad91f1e"
+      url: "https://pub.dev"
+    source: hosted
+    version: "0.2.1"
   petitparser:
     dependency: transitive
     description: