[FEATURE] Show active custom agent (subagent_type) in Claude App remote control view

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

When using the Claude App (iOS/Android) to remotely control a Claude Code session, the app gives no indication of which custom agent is currently active. If the session delegates to a custom agent (e.g. senior-engineer, business-analyst), this is entirely invisible in the remote UI — the app shows a generic in-progress state with no agent identity.

This makes it impossible to tell, at a glance, whether the orchestrator or a specific custom agent is doing the work. In the CLI TUI, the agent name/type is visible in the status bar and the agent tree view. The remote app has no equivalent.

Proposed Solution

When a custom agent (spawned via the Agent tool with a subagent_type) is active in a session, surface its name/type in the Claude App remote control view — for example, alongside the session status or as a subtitle under the current activity indicator. Something like:

Running: senior-engineer agent

This should stay in sync with the CLI status bar behaviour so the same information is available across surfaces.

Alternative Solutions

Currently there is no workaround — the app simply shows no agent context. The user must switch to the terminal to know which agent is running.

Priority

Medium - Would be very helpful

Feature Category

Interactive mode (TUI)

Use Case Example

1. I trigger a multi-step task from the CLI that spawns a senior-engineer agent.
2. I switch to my phone to monitor progress via remote control.
3. The app shows the session is active but gives no indication which agent is running.
4. With this feature, I'd see "Running: senior-engineer agent" and know exactly what stage the workflow is in without switching back to the terminal.

Additional Context

The CLI TUI already shows the active subagent name in the status bar and agent tree. The feature request is to bring that same visibility to the remote control surface in the Claude App, keeping both surfaces consistent.