Desktop app: session rename via set_session_title does not propagate to Remote Control (mobile) session list

## Description

Renaming a session's title in the Claude Code desktop app (via the `set_session_title` MCP tool exposed to the agent, same effect as the desktop UI rename) does not propagate to the session title shown in the Remote Control mobile view, even after manually pulling to refresh the session list or fully restarting the mobile app.

## Environment

- Platform: Windows 11, Claude Code desktop app (Code tab)
- Client showing stale title: Remote Control on Android (mobile "Code" app)
- Sessions affected: local sessions with `remoteControlActive: true` / `remoteControlState: "on"`

## Steps to reproduce

1. On desktop, have an active session that is connected via Remote Control (visible as "Connected · Remote control" on the mobile session list).
2. Rename the session (agent called `set_session_title`, equivalent to the desktop rename action). Desktop confirms success and `get_session` immediately reflects the new title.
3. On the phone, open the Remote Control session list ("Sessions" screen).
4. Pull to refresh the list, or fully close and reopen the mobile app.

## Expected

The mobile Remote Control session list shows the updated title.

## Actual

The mobile Remote Control session list keeps showing the **original/pre-rename title** indefinitely — confirmed after both a manual pull-to-refresh and a full app restart on the phone.

Example (titles genericized):
- Session A: renamed from `会话自动命名 Skill` → `20260918-新增-会话命名规则Skill` → later `20260918-基础设施-会话命名规则跨工具接入`. Desktop `get_session` correctly returns the latest title. Phone still shows `会话自动命名 Skill` (the very first, pre-rename title).
- Session B: renamed from `教材知识库同步与家教工作台优化` → `20260918-优化-知识库同步家教工作台`. Same divergence on the phone.

## Notes

- This looks like the desktop-side session title store and whatever backing data the Remote Control mobile session list reads from (likely a separate cloud-side session/bridge record) are not kept in sync when the title is changed from the desktop side.
- Not a caching issue on a short timer — the divergence persisted through both a manual refresh and a full mobile app restart.

Start by reproducing the rename through set_session_title with Remote Control enabled, then trace how the desktop title reaches the mobile Sessions list. Compare the desktop get_session result with the data returned after refresh and restart; done means the renamed title appears consistently on mobile.

Written by the indexing model from the issue text.