[BUG] "Fork" option missing from session context menu for Remote Control (server mode) sessions in Desktop app

Preflight Checklist

• [x] I have searched existing issues and this hasn't been reported yet
• [x] This is a single bug report (please file separate reports for different bugs)
• [x] I am using the latest version of Claude Code

What's Wrong?

The session-list context menu in Claude Desktop is missing the "Fork" option for sessions connected via Remote Control (server mode). Fork is present when right-clicking a locally-hosted Desktop session, but not when right-clicking an RC-hosted session.

What Should Happen?

Either Fork should work for RC-hosted sessions too (spawning a new session on the same remote host). Because I think there's no reason for this to be unavailable. We can create a new session with claude rc, why not fork?

Error Messages/Logs

Steps to Reproduce

1. On a remote machine, cd into a project and run claude remote-control (server mode).
2. Connect to that session from the Claude Desktop app (via session URL or the session list).
3. In Desktop's session list, right-click the RC-connected session -- note "Fork" is missing.
4. For comparison, right-click a purely local Desktop-hosted session -- "Fork" is present.

Claude Model

Not sure / Multiple models

Is this a regression?

I don't know

Last Working Version

_No response_

Claude Code Version

2.1.223 (Claude Code)

Platform

Other

Operating System

macOS

Terminal/Shell

Other

Additional Information

Claude Desktop version: 1.25927.0
Plan: Max
Platform/Terminal fields above are N/A -- this is a Desktop app GUI bug (session context menu), not a CLI/terminal session issue.