Inline autofill menu does not appear when Synology DSM automatically focuses password field

### Steps To Reproduce

Open the Synology DSM login page in Brave.
Click the username field and use Bitwarden to autofill the username.
Click the right-arrow button in the Synology login form to proceed to the password step.
Synology displays the password field and automatically places focus in it; the text cursor is already visible in the password field.
Observe that the Bitwarden inline autofill menu does not appear.
Click somewhere outside the password field.
Click back inside the password field.
Observe that the Bitwarden inline autofill menu now appears.

### Expected Result

When Synology displays and automatically focuses the password field, the Bitwarden inline autofill menu should be available immediately. I should not need to blur the field and manually focus it again.

### Actual Result

When Synology automatically focuses the newly displayed password field, the Bitwarden inline autofill menu does not appear.

The menu only appears after I click outside the password field and then click back into it.

Bitwarden keyboard autofill using Ctrl+Shift+L works immediately while the password field is initially focused.

### Screenshots or Videos

_No response_

### Additional Context

The Synology DSM login is a two-step login flow. After the username is autofilled, clicking Synology's right-arrow button causes the password field to be dynamically displayed and automatically focused.

The important detail is that the password field already has focus and the cursor is visible when it first appears. The Bitwarden inline autofill menu does not appear at that point.

A subsequent manual blur/focus causes the inline autofill menu to appear.

Ctrl+Shift+L works immediately while the password field is initially focused, so Bitwarden is able to identify and autofill the credential. This appears to be specific to the inline autofill menu not responding to the dynamically created and automatically focused password field.

I have also observed the same general behavior on other websites: on some login pages, when a password field receives focus automatically or is dynamically displayed, the Bitwarden inline menu does not appear until I click outside the field and then click back into it.

Synology DSM provides a consistent and easily reproducible example of the issue.

The issue occurs in Brave with the Bitwarden browser extension.

### Operating System

Windows

### Operating System Version

Win 11, Zorin 18

### Web Browser

Brave

### Browser Version

Brave 1.93.134, Firefox 153.0.4, LibreWolf 153.0.3-1

### Environment Versions

Version: 2026.7.0
SDK: 'main (3e060f8)'
Server version: 2026.7.2

### 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 with the Synology DSM two-step login flow, focusing on the dynamically displayed and automatically focused password field. Trace how the inline autofill menu responds to automatic focus versus a manual blur/focus cycle. Done means the menu appears immediately when the password field is first focused, while existing keyboard autofill continues to work.

Written by the indexing model from the issue text.