Browserbase MCP connector: `start` tool has no way to pass browserSettings.context (persisted login)

This concerns the Browserbase MCP connector as exposed in the Claude app / Cowork (not the CLI), so the CLI-specific fields in any template don't apply.

What's missing

Browserbase's own API supports persisted browser contexts (saved cookies/login state) via POST /v1/sessions with browserSettings.context = {id, persist: true}. This is the documented way to reuse a logged-in session (e.g. a site login done once via the Live View debug URL) across future sessions.

The mcp__Browserbase__start tool exposed to Claude only accepts an optional sessionId parameter (to reuse/reattach to an existing live session). It has no parameter for browserSettings.context, so there's no way to create a new session that attaches a persisted context.

Repro

1. Create a Browserbase context via the Browserbase dashboard/API and log into a site manually through the Live View (e.g. LinkedIn via Google SSO).
2. In a Claude session with the Browserbase connector, call start (no params, or with an unrelated sessionId).
3. Navigate to the site that should be logged in.
4. Result: redirected to the login page. The new session is a blank browser profile; the persisted context was never attached because start has no way to request it.

Impact

Any workflow relying on persisted logins (the main reason to use Browserbase contexts at all — testing logged-in flows, automating an already-authenticated account) is unusable through this connector. The only way to reuse a login is to keep the same live session alive indefinitely via sessionId reattachment, which doesn't survive the session ending/timing out and isn't how Browserbase's own context feature is meant to be used.

Ask

Add an optional context (or contextId + persist) parameter to the start tool that maps to Browserbase's browserSettings.context, so a new session can attach a previously-created persisted context.