Fix Android/iOS VoIP Calling + Push Notification Routing (Flutter App)

I have a mobile app built in Flutter (Android & iOS). The app includes VoIP-style calling features using Firebase and flutter_callkit_incoming.

The core logic is working: the user receives push notifications and call events, but the call screen no longer appears on lock screen, and tapping Accept only opens the home screen.

I need a Flutter expert to:

Fix call intent routing on push (both Android & iOS)

Ensure the call screen appears properly (even in background/locked state)

Handle FirebaseMessaging.onMessage, onMessageOpenedApp, and getInitialMessage

Help debug and test with FCM payloads

You will get access to:

The full repo (mobile & backend)

Firebase credentials and serviceAccountKey

Existing AWS Lambda FCM logic

This is a repair/debug task, not a fresh build.

Please only apply if you already worked with flutter_callkit_incoming or similar VoIP packages and Firebase push routing.