Render statusline output in Remote Control web/mobile UI

Problem

When using Claude Code via Remote Control (web/mobile), the user-configured statusline is not displayed. The statusline is a critical signal surface in the terminal — it commonly shows context %, model, working directory, branch, cost, and custom indicators powered by user hooks. None of this reaches Remote Control users.

This was originally raised in #35483 ("Context usage not visible during remote control sessions"), which was closed as not planned by stale-bot after 7 days with no triage or response from a maintainer. Refiling per the closing comment's invitation, and broadening from "context %" to the more general fix: just pipe the existing statusLine command's stdout to the Remote Control UI.

Impact

• Users on mobile/remote cannot gauge when to checkpoint or start a fresh session — by the time the model starts degrading, it's too late for a clean handoff.
• Hooks that surface state via the statusline (context alerts, cost trackers, custom indicators) are silently degraded on remote — the data is being computed but never reaches the user.
• Power users lose every custom workflow signal they've invested in building.
• This is the only remaining signal-loss surface for Remote Control. Related work in #36096, #31840, and #54981 is flowing Remote Control state into the statusline JSON; nothing flows the statusline out to remote.

Suggested approach

Pipe the user-configured statusLine command's stdout (already computed for the terminal) to the Remote Control client and render it as a header or footer banner. No new data sources, no new computation — just surface what's already there.

This feels like a small change with outsized value: it makes every existing user hook and statusline customization work uniformly across surfaces.

Related

• #35483 (closed stale, original request — never triaged)
• #18456 (analogous gap in the VSCode extension)
• #36096, #31840, #54981 (open, inverse direction — RC state into statusline)