앱 브리지 메소드명 변경: updateMessageCount -> updateAlarmCount

- updateMessageCount를 updateAlarmCount로 변경
- BridgeMessageType.UPDATE_MESSAGE_COUNT를 UPDATE_ALARM_COUNT로 변경
- header에서 알림 카운트 조회 시 네이티브 앱에 카운트 업데이트 적용

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

src/widgets/navigation/header.tsx

@@ -2,7 +2,7 @@ import { useNavigate } from '@/shared/lib/hooks/use-navigate';
 import { Menu } from '@/shared/ui/menu';
 import { IMAGE_ROOT } from '@/shared/constants/common';
 import { PATHS } from '@/shared/constants/paths';
-import { 
+import {
   HeaderType,
   HeaderNavigationProps
 } from '@/entities/common/model/types';
@@ -10,6 +10,7 @@ import { useStore } from '@/shared/model/store';
 import { ChangeEvent, useEffect, useState } from 'react';
 import { AppAlarmUnreadCountParams, AppAlarmUnreadCountResponse, MERCHANT_ADMIN_APP } from '@/entities/alarm/model/types';
 import { useAppAlarmUnreadCountMutation } from '@/entities/alarm/api/use-app-alarm-unread-count-mutation';
+import { appBridge } from '@/utils/appBridge';
 
 export const HeaderNavigation = ({ 
   onBack, 
@@ -63,6 +64,12 @@ export const HeaderNavigation = ({
       };
       appAlarmUnreadCount(params).then((rs: AppAlarmUnreadCountResponse) => {
         setUnreadCount(rs.unreadCount);
+        // 네이티브 앱에 알림 카운트 업데이트
+        if(appBridge.isNativeEnvironment()){
+          appBridge.updateAlarmCount(rs.unreadCount).catch((error) => {
+            console.error('Failed to update alarm count:', error);
+          });
+        }
       });
     }
   };