[FEATURE] Support dispatching tasks from a second desktop/browser client, not only from mobile

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

Currently, Dispatch requires both the Claude Desktop app and the Claude mobile app Claude — the only way to remotely assign tasks to a host desktop is through a paired smartphone. There is no way to dispatch tasks from a second computer, a browser tab, or any non-mobile client.
This is a real limitation for users who work across multiple machines — for example, a personal laptop and a work laptop — and don't have their phone nearby or prefer not to use it as an intermediary. It also creates friction in headless or corporate environments where mobile devices aren't part of the workflow.

Proposed Solution

Extend the Dispatch pairing model to support a second desktop client (or claude.ai in browser) as a valid dispatch origin, in addition to the existing mobile pairing. The host machine would remain the executor; the second client would just act as the remote input surface, the same role the phone plays today.
This could be exposed as:

A second paired Claude Desktop instance (similar to the current mobile QR pairing flow)
A claude.ai browser session authenticated to the same account
A CLI entry point (claude dispatch) for power users

Alternative Solutions

_No response_

Priority

High - Significant impact on productivity

Feature Category

CLI commands and flags

Use Case Example

_No response_

Additional Context

_No response_