Can't login to Bitwarden while HTTP basic authentication prompt is displayed

### Steps To Reproduce

1. Open browser with Bitwarden extension
2. Logout from Bitwarden extension
3. Open any webpage with basic HTTP authentication, for instance https://testpages.eviltester.com/pages/auth/basic-auth/basic-auth-results.html
4. Try to login to Bitwarden extension while the basic HTTP authentication prompt is displayed

### Expected Result

To be able to login.

### Actual Result

Bitwarden extension asks for credentials but nothing happens.

### Screenshots or Videos

_No response_

### Additional Context

Happens with any basic HTTP prompt. In this scenario the user must close all tabs with HTTP basic auth prompts, login to Bitwarden, and finally use Bitwarden extension successfully.
Note that this also happens when the tab with the basic HTTP auth prompt is not the active one.

### Operating System

Windows

### Operating System Version

Windows 11 Enterprise 10.0.26200 build 26200

### Web Browser

Chrome

### Browser Version

151.0.7922.72 64-bit

### Environment Versions

Version: 2026.7.0

SDK: 'main (3e060f8)'

Server version: 2026.7.1

### 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.

The issue names no repository files or tests. First reproduce the browser-extension login flow while a Chrome HTTP basic-auth prompt is open, then trace the affected login entry point; done means Bitwarden login succeeds without closing any basic-auth prompts, including prompts in inactive tabs.

Written by the indexing model from the issue text.