Phase 3.1 WS2: FCM fallback Flutter + CC, unread badges, dynamic ping

- Control center: add mitra ping config UI (require ping toggle + interval)
- Mitra app StatusNotifier: honor require_ping and ping_interval_seconds
  from API; skip heartbeat when ping not required
- Both apps: update notification services for FCM deep-linking
  - mitra_app: handle chat_request (open_accept), session_closing
  - client_app: handle session_closing, paired
- Unread badge providers:
  - mitra_app: UnreadSessions provider (polls active-with-unread, badge
    on active sessions button)
  - client_app: UnreadCount provider (polls active-with-unread, badge
    on _ActiveSessionCard)

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

client_app/lib/features/home/home_screen.dart

@@ -3,6 +3,7 @@ import 'package:flutter_riverpod/flutter_riverpod.dart';
 import 'package:go_router/go_router.dart';
 import '../../core/auth/auth_notifier.dart';
 import '../../core/api/api_client_provider.dart';
+import '../../core/chat/unread_notifier.dart';
 import '../../core/pairing/pairing_notifier.dart';
 import '../chat/widgets/pricing_bottom_sheet.dart';
 
@@ -133,15 +134,16 @@ class _HomeScreenState extends ConsumerState<HomeScreen> with WidgetsBindingObse
   }
 }
 
-class _ActiveSessionCard extends StatelessWidget {
+class _ActiveSessionCard extends ConsumerWidget {
   final Map<String, dynamic> session;
   final VoidCallback onTap;
 
   const _ActiveSessionCard({required this.session, required this.onTap});
 
   @override
-  Widget build(BuildContext context) {
+  Widget build(BuildContext context, WidgetRef ref) {
     final mitraName = session['mitra_display_name'] as String? ?? 'Bestie';
+    final unreadCount = ref.watch(unreadCountProvider);
 
     return Card(
       elevation: 2,
@@ -152,9 +154,13 @@ class _ActiveSessionCard extends StatelessWidget {
           padding: const EdgeInsets.all(20),
           child: Row(
             children: [
-              const CircleAvatar(
-                backgroundColor: Colors.green,
-                child: Icon(Icons.chat, color: Colors.white),
+              Badge(
+                isLabelVisible: unreadCount > 0,
+                label: Text('$unreadCount'),
+                child: const CircleAvatar(
+                  backgroundColor: Colors.green,
+                  child: Icon(Icons.chat, color: Colors.white),
+                ),
               ),
               const SizedBox(width: 16),
               Expanded(