[FEATURE] Remote Control: option to send mobile push notifications even while the terminal is focused

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

Remote Control intentionally skips mobile push notifications while the connected terminal is focused/typed-in (documented behavior since v2.1.181). But "terminal is focused" doesn't always mean "I'm actually at the machine" — e.g. I leave the terminal window frontmost on my Mac and step away, checking progress from my phone. In that case I never get a push, even though I'm not actually watching the terminal.

Proposed Solution

Add an opt-in toggle (alongside the existing /config options "Push when Claude decides" and "Push when actions required") to always send Remote Control mobile pushes regardless of local terminal focus state.

Alternative Solutions

CLAUDE_CLIENT_PRESENCE_FILE exists, but it only widens suppression (skips pushes even in other windows while "present"), which is the opposite of what I need.

Priority

Medium - Would be very helpful

Feature Category

Configuration and settings

Use Case Example

1. I kick off a long task in Claude Code on my Mac and leave the terminal window open/frontmost.
2. I walk away and check progress from my phone via Remote Control.
3. Because the terminal is still "focused" on the Mac, no push arrives, so I have no way to know the task finished without manually reopening the session.

Additional Context

Related: #79617 (presence-aware notification control, closed/stale) discusses similar presence-check plumbing for terminal-channel notifications.