[FEATURE] Remote Control: "Send now" on a queued message should not kill running background agents and tasks

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 I drive a session through Remote Control (claude.ai web and mobile) and the session is busy, my message gets queued. Tapping "Send now" on the queued message interrupts the turn, and that interrupt also stops every background agent and background task that is running at the time. So pushing one short message through right away throws away the work of all running agents, often a long test run or several agents in parallel.

Proposed Solution

"Send now" on a queued message should deliver the message right away but leave running background agents and background tasks alone. Only Stop should end them. The terminal already works this way since 2.1.281: its send-now key moves running work to the background instead of cancelling it (https://code.claude.com/docs/en/interactive-mode#when-claude-code-sends-what-you-queued). Remote Control's "Send now" should do the same.

Alternative Solutions

Leaving the message queued avoids the kill, because a queued message is handed over at the next tool-call boundary without an interrupt. But it can lag several tool calls behind (#92899), which is why "Send now" exists. A setting to choose between queueing and interrupting (#87656) would also help, but the default for "Send now" should not kill background work.

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

1. From the phone, the session is running four background agents that each run a long test suite.
2. While the main turn is busy, I type a correction and it gets queued.
3. I tap "Send now" so the correction reaches the session immediately.
4. Today: all four agents are killed and their work is lost. Expected: the correction is delivered and the four agents keep running.

Additional Context

Sessions run through claude remote-control, currently on 2.1.280. Related: #95211 (Stop from Remote Control killed four background agents), #78151 and #83024 (an interrupt silently kills background agents), #87656 (setting for queue vs interrupt), #92899 (steering a running task is slow or destructive).