[FEATURE] Desktop Browser pane: support the Bitwarden extension

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Claude Desktop 2.1.288 on Linux. The Browser pane can't run browser extensions, so I can't use Bitwarden in it. The app also blocks the Chrome Web Store there: opening it returns "can't load in the Browser pane, since the Chrome Web Store only works in a full browser."

The pane is the only browser surface that runs per session and in parallel, which makes it the one I want to use. Without Bitwarden, every site I log into means typing a password by hand. The alternative is Claude in Chrome, which drives my everyday browser and handles one session at a time.

Proposed Solution

Let users add the Bitwarden extension to the Browser pane's profile, off by default and switched on in settings.

1. Install it from its Chrome Web Store page, or from an unpacked folder.
2. Bitwarden acts for the user only. The agent can't read its vault, popup or storage, and an autofill happens outside the model's view.
3. The vault stays locked until I unlock it in the pane myself.

Electron already has session.loadExtension, with partial coverage of the Chrome extension APIs. Bitwarden's autofill and popup are enough. Other extensions can come later.

Alternative Solutions

1. Claude in Chrome with Bitwarden in my main browser. It works, but it takes over my everyday browser and serializes sessions.
2. A native credential handoff where a password manager fills fields and the model never sees values (#88165, #96762, #83380). That would also solve this, though Bitwarden in the pane is a smaller step.
3. Signing in to each site by hand in the pane. That's what I do today.

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

I run several Claude Code sessions at once, each with its own Browser pane, against dashboards I'm logged into (Vercel, Cloudflare, Webflow). With Bitwarden in the pane I would unlock once, fill each login myself, and let the sessions work in parallel without touching my main browser.

Additional Context

The Codex app's integrated browser doesn't have this either (openai/codex#32081 asks for 1Password there). Codex's answer so far is a Chrome extension that drives your real browser, which is how Claude in Chrome works.

Related: #87764 (no way to bring existing login state into the pane), #87853 (closed: no system password manager in the pane).