[BUG] Agent Teams: a teammate in its own iTerm2/tmux pane shows "manual mode" and a timer that never stops in the lead's teammate view

What's wrong?

When a teammate runs in its own terminal pane (teammateMode resolves to the iTerm2 or tmux backend), switching to it in the lead (Viewing @<name>) shows state that is wrong and never updates:

1. Wrong permission mode. The footer says manual mode on, although the teammate process was launched with --permission-mode auto and its own pane shows auto mode on.
2. The timer never stops. The spinner and the task row keep counting (Pouncing… (8m 0s)) after the teammate has finished its turn and sent its idle_notification. The teammate's own pane shows done.
3. The view is empty apart from the spawn prompt (related to #82687 and #98550, but with a different cause; see below).

Cause (read from the 2.1.284 bundle)

For a pane teammate, the lead registers an in_process_teammate task record once, at spawn, with hard-coded values:

permissionMode: planModeRequired ? "plan" : "default",
isIdle: false,

• "default" is shown as "Manual" in the mode table, whatever --permission-mode was passed to the pane process.
• isIdle is set to true only by the in-process runner, which never runs for a pane teammate, so the record stays "running" forever. The idle_notification the lead receives does not update it.
• No messages are ever added to the record, because the teammate's turns happen in another process. The view therefore shows only the spawn prompt.

This differs from #98550, where an in-process teammate's messages are in memory but trimmed to 50, and then to one message on completion. Fixing that would not help the pane case.

Expected

The lead's view of a pane teammate reflects the live process:

• the permission mode it was actually launched with,
• idle or working, using the idle_notification the lead already receives,
• either its transcript or a clear pointer to the pane that holds it.

Environment

Claude Code 2.1.284, macOS, iTerm2 split-pane backend, CLAUDE_CODE_EXPERIMENTAL_AGENT_TEAMS=1.

Related: #98550, #82687, #75190