Browser extension on Arc Browser(Chromium) forcing me to refresh the page to authenticate

### Steps To Reproduce

1. Go to any page with Arc browser
2. Click on the bitwarden extension icon
3. You will see a popup on the top left with cut off text "your updated settings on this site" with 2 action buttons "Reload" and "Cancel".

### Expected Result

I should be able to authenticate the extension without being forced to reload the page.

### Actual Result

The popup forces you to reload the page.
After the reload it'll work as intended the next time you click on the extension icon.

### Screenshots or Videos

### Additional Context

_No response_

### Operating System

macOS

### Operating System Version

Sonoma 14.5

### Web Browser

Chrome

### Browser Version

Arc version: 1.142.1, Chromium Engine Version: 147.0.7727.56

### Environment Versions

Extension Version: 2026.3.0
Server Version: 2025.7.0

### 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 issue in the Bitwarden browser extension using Arc on macOS with the reported versions, then trace the flow opened by the extension icon during authentication. Check how the popup handles the "updated settings on this site" prompt. Done means authentication proceeds without forcing a page reload, while the existing post-reload behavior remains intact.

Written by the indexing model from the issue text.