fix(background-agent): preserve non-sensitive timing in archived tasks

Carry queuedAt and startedAt in sanitized archive snapshots so post-cleanup task output duration remains accurate without retaining prompt content.

Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)

Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>

src/features/background-agent/manager.test.ts

@@ -6017,6 +6017,7 @@ describe("BackgroundManager regression fixes - resume and aborted notification",
     const archivedTask = manager.getTask(task.id)
     expect(archivedTask?.sessionId).toBe(task.sessionId)
     expect(archivedTask?.prompt).toBe("[redacted]")
+    expect(archivedTask?.startedAt).toEqual(task.startedAt)
 
     manager.shutdown()
   })

src/features/background-agent/manager.ts

@@ -383,6 +383,8 @@ export class BackgroundManager {
       agent: task.agent,
       sessionId: task.sessionId,
       status: task.status,
+      queuedAt: task.queuedAt,
+      startedAt: task.startedAt,
       completedAt: task.completedAt,
       model: task.model,
       error: task.error,