Bitwarden extension for Safari private windows

### Steps To Reproduce

1. Go to 'Safari' menu
2. Click on 'Safari Extensions...'
3. Scroll down to 'Bitwarden Password Manager'
4. Click on 'Allow in Private Browsing'
5. Use biometrics to confirm selection

### Expected Result

Expected the Bitwarden Password Manager (BPM) icon to appear in the Safari private window toolbar when confirming the selection via biometrics.

### Actual Result

The BPM icon does not appear in the Safari private window toolbar; the toolbar remains unchanged.

### Screenshots or Videos

_No response_

### Additional Context

Whether confirming the 'Allow in Private Windows' selection via biometrics or with the current user's password, the result should be the same: The BPM icon should appear in the Safari private window toolbar. However, the BPM icon only appears when confirming the selection via the current user's password.

### Operating System

macOS

### Operating System Version

Tahoe 26.1

### Web Browser

Safari

### Browser Version

26.1 (21622.2.11.11.9)

### Environment Versions

Version: 2025.11.0

SDK: 'main (8ef7951)'

Server version: 2025.11.0

### Issue Tracking Info

- [x] I understand that work is tracked outside of Github. A PR will be linked to this issue should one be opened to address it, but Bitwarden doesn't use fields like "assigned", "milestone", or "project" to track progress.

Start by reproducing the Safari 26.1 private-window flow on macOS, comparing biometric confirmation with password confirmation. Trace the browser-extension path for enabling private browsing and verify that the Bitwarden Password Manager icon appears in the private-window toolbar for both methods.

Written by the indexing model from the issue text.