[FEATURE] Remote Control: Push notifications on iOS when permission approval is needed

Description

When using Remote Control from the iOS Claude app, there's no push notification when the CLI session needs permission approval (e.g., tool calls waiting for user consent). You have to keep the app open and actively watch the session to notice when it's blocked on a permission prompt.

Expected Behavior

The iOS Claude app should send a push notification when a Remote Control session is waiting for permission approval, so the user can step away and be alerted when their input is needed — similar to how CI/CD systems notify on pipeline approval gates.

Current Behavior

No notification is sent. The session silently blocks until you happen to check the app, which can waste significant time if you're away from your phone.

Why This Matters

Remote Control is most useful when you can kick off a task and walk away. Without notifications, you have to babysit the session to catch permission prompts, which defeats the purpose of remote control.

Environment

• Platform: iOS (Claude app)
• Feature: Remote Control