refactor(background-agent): extract inline logic from manager.ts into focused modules

Extract 5 concerns from BackgroundManager into dedicated modules:
- error-classifier.ts: enhance with extractErrorName, extractErrorMessage, getSessionErrorMessage, isRecord
- fallback-retry-handler.ts: standalone tryFallbackRetry with full retry logic
- process-cleanup.ts: registerManagerForCleanup/unregisterManagerForCleanup
- compaction-aware-message-resolver.ts: isCompactionAgent/findNearestMessageExcludingCompaction
- Delete notification-builder.ts (duplicate of background-task-notification-template.ts)

Manager.ts method bodies now delegate to extracted modules.
Wire duration-formatter.ts and task-poller.ts (existing but unused).

manager.ts: 2036 -> 1647 LOC (19% reduction).
All 161 existing tests pass unchanged.

src/features/background-agent/compaction-aware-message-resolver.ts

@@ -0,0 +1,57 @@
+import { readdirSync, readFileSync } from "node:fs"
+import { join } from "node:path"
+import type { StoredMessage } from "../hook-message-injector"
+
+export function isCompactionAgent(agent: string | undefined): boolean {
+  return agent?.trim().toLowerCase() === "compaction"
+}
+
+function hasFullAgentAndModel(message: StoredMessage): boolean {
+  return !!message.agent &&
+    !isCompactionAgent(message.agent) &&
+    !!message.model?.providerID &&
+    !!message.model?.modelID
+}
+
+function hasPartialAgentOrModel(message: StoredMessage): boolean {
+  const hasAgent = !!message.agent && !isCompactionAgent(message.agent)
+  const hasModel = !!message.model?.providerID && !!message.model?.modelID
+  return hasAgent || hasModel
+}
+
+export function findNearestMessageExcludingCompaction(messageDir: string): StoredMessage | null {
+  try {
+    const files = readdirSync(messageDir)
+      .filter((name) => name.endsWith(".json"))
+      .sort()
+      .reverse()
+
+    for (const file of files) {
+      try {
+        const content = readFileSync(join(messageDir, file), "utf-8")
+        const parsed = JSON.parse(content) as StoredMessage
+        if (hasFullAgentAndModel(parsed)) {
+          return parsed
+        }
+      } catch {
+        continue
+      }
+    }
+
+    for (const file of files) {
+      try {
+        const content = readFileSync(join(messageDir, file), "utf-8")
+        const parsed = JSON.parse(content) as StoredMessage
+        if (hasPartialAgentOrModel(parsed)) {
+          return parsed
+        }
+      } catch {
+        continue
+      }
+    }
+  } catch {
+    return null
+  }
+
+  return null
+}