[PM-27213] Auto-Fill Not Working in Edge Browser on iPadOS via Bitwarden App

### Steps To Reproduce

1. Install the Bitwarden app on an iPad running iPadOS.
2. Log in to the Bitwarden app using Bitwarden Account.
3. In the Bitwarden app, go to Settings → Account Security → Unlock with Face ID, and enable Face ID unlocking.
4. Connect a physical keyboard to the iPad.
5. Open the Microsoft Edge browser on the iPad.
6. Navigate to a website that has saved login credentials in Bitwarden.
7. Trigger the auto-fill function using Bitwarden’s integration in the Edge browser with Face ID unlocking.

### Expected Result

Auto-fill should populate the account name and password fields, just as it works in other apps.

### Actual Result

The account name and password fields are not automatically populated in the Edge browser when a physical keyboard is connected.

### Screenshots or Videos

_No response_

### Additional Context

_No response_

### Build Version

2025.10.0 (2645)

### What server are you connecting to?

US

### Self-host Server Version

_No response_

### Environment Details

© Bitwarden Inc. 2015–2025

📝 Bitwarden 2025.10.0 (2645)
📦 Bundle: com.8bit.bitwarden
📱 Device: iPad8,1
🍏 System: iPadOS 18.6.2
🧱 Commit: https://github.com/bitwarden/ios/commit/277efe6d95f354ded3b73f1c5ff7f16d82c9d962
💻 Build Source: bitwarden/ios/actions/runs/18227666097/attempts/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.

Start by reproducing the issue on the listed iPadOS, iPad, Edge, keyboard, and Bitwarden build combination, then trace the iPadOS autofill integration used by the Bitwarden app. Done means the saved account name and password populate in Edge with Face ID unlocking while a physical keyboard is connected.

Written by the indexing model from the issue text.