[Feature] Let an SDK / -p host declare its session attended, so background commands aren't stopped at 30 minutes (as in VS Code since 2.1.288)

Summary

Since 2.1.288, the background command time limit applies only to unattended sessions (-p, Agent SDK, CI, cloud); terminal, desktop and VS Code sessions have none. An SDK host has no way to say that its session is attended, so a chat UI built on the Agent SDK / -p --input-format stream-json still stops every run_in_background command at 30 minutes (2 h at most), even with a person watching the chat.

Current behaviour (read from the 2.1.291 bundle)

• The limit applies when the session is non-interactive and is neither the desktop app nor CLAUDE_CODE_ENTRYPOINT=claude-vscode.
• The only other way out is the host object's disableBackgroundDeadline(), which only claude mcp serve calls. No setting, environment variable or initialize option reaches it.
• CLAUDE_CODE_SESSION_ATTENDED exists (the CLI sets it for its child processes), but the deadline does not read it.
• So a third-party editor integration (ours is a VS Code extension that hosts Claude Code chats in its own webview) can only:
• spoof the native panel's entrypoint, which also changes everything else that entrypoint means (telemetry, IDE integration checks); or
• raise BASH_MAX_TIMEOUT_MS and fill in a long timeout on every background call.

We ship the second for now. It also raises the foreground maximum, and it can't cover a foreground command the CLI moves to the background later (Ctrl+B or auto-background), since no new tool call happens for that move.

Request

A supported way for an SDK host to declare its session attended, so it gets the same background-command behaviour as the VS Code panel. Any of these would work for us:

1. Honour CLAUDE_CODE_SESSION_ATTENDED=1 in the session's own environment for the deadline check.
2. An initialize control-request option (for example attended: true) or an SDK option that calls disableBackgroundDeadline().
3. A setting such as backgroundTaskTimeout: "none".

Related: #80372 (background tasks stopped in long sessions, before 2.1.288).

Environment: Claude Code 2.1.291, Windows 11 and Linux, claude -p --input-format stream-json --output-format stream-json, entrypoint sdk-cli.