[Feature Request] Background task execution without conversation blocking in VSCode Claude extension

Bug Description
Title: Feature Request: Run tasks in the background without blocking the conversation

Body:

Currently, long-running tasks block the conversation until they finish. I'd like a way to kick off a task in the background and keep the main conversation free for other work 0 similar to how the CLI supports background agent execution.

Use case: Start a long refactor or test run in the background, continue asking questions or doing other work in the same session.