[BUG] Scheduled-task sessions never register with Remote Control / no session appears in mobile app

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

When a scheduled task (created via the scheduled-tasks feature, whether recurring cron or a one-time fireAt) executes, the task runs successfully — list_scheduled_tasks shows lastRunAt updating correctly — but the resulting session never appears in the Remote Control session list on the Claude mobile app (Code tab). Any PushNotification calls made during the run are also never delivered to the phone. This happens even with "Connect new sessions to Remote Control" enabled in Desktop app settings.

### What Should Happen?

The scheduled-task session should register with Remote Control the same way a normal interactive session does, appear in the mobile app's session list, and any push notifications sent during the run should reach the phone.

### Error Messages/Logs

```shell
No error is surfaced anywhere. The task simply completes silently with no corresponding Remote Control session ever created.
```

### Steps to Reproduce

1. In Claude Desktop app, enable Settings > Claude Code > Local sessions > "Connect new sessions to Remote Control"
2. Create a scheduled task (via the scheduled-tasks feature) with a one-time fireAt a few minutes in the future, with a prompt that just calls the PushNotification tool with a test message
3. Wait for the fireAt time to pass, then check list_scheduled_tasks — lastRunAt is updated, confirming the task executed
4. Open the Claude mobile app > Code tab > session list
5. Observe: no new session appears for this run, and no push notification is received, even though the task executed successfully

Additionally, two long-running production scheduled tasks show the same pattern in real usage: their last visible mobile-side session entries stop around 2026-09-03/04, while lastRunAt kept updating daily through 2026-09-08 — indicating the session/push registration silently stopped working around that date while the task itself kept running.

### Claude Model

None

### Is this a regression?

Yes, this worked in a previous version

### Last Working Version

Unknown exact version — was working normally until approximately 2026-09-03/04

### Claude Code Version

2.1.266 (Claude Code)

### Platform

Anthropic API

### Operating System

Windows

### Terminal/Shell

PowerShell

### Additional Information

Issue occurs via Claude Desktop App on Windows, not terminal-specific

No files or tests are named. Start by reproducing the one-time scheduled task and trace scheduled-task execution through Remote Control session registration and PushNotification delivery; done means the run creates a mobile-visible session and delivers its notification, while interactive sessions remain unaffected.

Written by the indexing model from the issue text.