[Bug] Cannot paste 2FA code when logging in to Bitwarden in Safari

### Bitwarden Beta

- [X] I'm using the new native Bitwarden Beta app and I'm aware that legacy .NET app bugs should be reported in [bitwarden/mobile](https://github.com/bitwarden/mobile)

### Steps To Reproduce

1. Go to Settings -> Passwords -> Password Options.
2. Turn off Bitwarden if it is on.
3. Turn off AutoFill Passwords and Passkeys.
4. Turn AutoFill Passwords and Passkeys back on.
5. Turn Bitwarden on.
6. You will be asked to log in to Bitwarden, and will see an option to Paste the 2FA code.
7. Open Safari and browse a website where you have a password stored for in Bitwarden.
8. You will be asked to log in to Bitwarden again.
9. This time, you will not see a Paste button.

### Expected Result

Have both Paste and AutoFill (similar to when logging in to the Bitwarden app in iOS).

### Actual Result

No Paste button.

### Screenshots or Videos


### Build Version

2024.7.3

### Environment Details

iPhone 15 Pro Max
17.5.1

### 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 login flow on the listed iPhone and iOS version, following the Safari steps and comparing it with the Bitwarden app login. Trace the iOS Safari AutoFill login path to determine why the Paste option is absent. Done means Safari login offers both Paste and AutoFill, matching the expected app behavior.

Written by the indexing model from the issue text.