Badge notifications can be lost when clearing and immediately setting a new badge value

### Describe the bug

When a badge is cleared and a new badge is posted immediately afterwards, the new badge may not appear.

## Additional observations

* Reproduced with both `Microsoft.Windows.BadgeNotifications.BadgeNotificationManager` and `Windows.UI.Notifications.BadgeUpdater`.
* Adding a delay of approximately 200 ms between the clear and update operations consistently works.
* A delay of 100 ms is insufficient on the tested system.
* This suggests a possible timing/ordering issue in the underlying badge notification pipeline rather than in a specific API surface.

### Steps to reproduce the bug

``` csharp
var badgeNotificationManager = BadgeNotificationManager.Current;

badgeNotificationManager.SetBadgeAsCount(100); // shows "99+"
badgeNotificationManager.SetBadgeAsCount(0); // clears
badgeNotificationManager.SetBadgeAsCount(99); // expected: 99
// actual: no badge
```

## Workaround
``` csharp
badgeNotificationManager.SetBadgeAsCount(0);
await Task.Delay(200);
badgeNotificationManager.SetBadgeAsCount(99);
```

### Expected behavior

The badge should immediately update from "99+" to "99" without requiring a delay.

### Screenshots

_No response_

### NuGet package version

2.2.0

### Packaging type

Packaged (MSIX)

### Windows version

_No response_

### IDE

Other

### Additional context

* Reproduced with Windows App SDK 2.2.
* OS build: **26200.8655**
* IDE: Visual Studio 2026

Start by reproducing the transition through Microsoft.Windows.BadgeNotifications.BadgeNotificationManager and compare it with Windows.UI.Notifications.BadgeUpdater. Trace the badge notification pipeline to determine why clearing followed immediately by setting 99 can lose the update; done means the badge changes from 99+ to 99 without a delay on the reported Windows App SDK version and OS build.

Written by the indexing model from the issue text.