[PM-43138] fix(browser): show save login after generated password

[tmsteph]

🎟️ Tracking

Fixes #18167

📔 Objective

When a password is generated from the inline autofill menu, the generated value is recorded, but the inline list is closed by the resulting input event. Because the password field can remain focused, there may be no later focus event to reopen the inline menu and expose the existing Save to Bitwarden view.

After generatedPasswordFilled is recorded, request openAutofillInlineMenu again. This reuses the existing filled-field logic and save-login UI rather than introducing a new save path or automatically writing to the vault.

The regression assertion verifies that the generated-password handler first records generatedPasswordFilled and then requests the inline menu to reopen.

Validation performed locally:

• git apply --check
• git diff --check
• patch verified against current bitwarden/clients main

I did not run the full browser test suite locally because installing the monorepo dependency set exhausted the available worker resources. CI can provide the full project validation.

📸 Screenshots

Not included: this does not add or restyle UI; it reopens Bitwarden's existing filled-field Save to Bitwarden inline-menu state after generated-password fill.

[CLAassistant]

All committers have signed the CLA.

[bitwarden-bot]

Thank you for your contribution! We've added this to our internal tracking system for review.
ID: PM-43138
Link: https://bitwarden.atlassian.net/browse/PM-43138

Details on our contribution process can be found here: https://contributing.bitwarden.com/contributing/pull-requests/community-pr-process.