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/error-classifier.ts

@@ -1,3 +1,7 @@
+export function isRecord(value: unknown): value is Record<string, unknown> {
+  return typeof value === "object" && value !== null
+}
+
 export function isAbortedSessionError(error: unknown): boolean {
   const message = getErrorText(error)
   return message.toLowerCase().includes("aborted")
@@ -19,3 +23,61 @@ export function getErrorText(error: unknown): string {
   }
   return ""
 }
+
+export function extractErrorName(error: unknown): string | undefined {
+  if (isRecord(error) && typeof error["name"] === "string") return error["name"]
+  if (error instanceof Error) return error.name
+  return undefined
+}
+
+export function extractErrorMessage(error: unknown): string | undefined {
+  if (!error) return undefined
+  if (typeof error === "string") return error
+  if (error instanceof Error) return error.message
+
+  if (isRecord(error)) {
+    const dataRaw = error["data"]
+    const candidates: unknown[] = [
+      error,
+      dataRaw,
+      error["error"],
+      isRecord(dataRaw) ? (dataRaw as Record<string, unknown>)["error"] : undefined,
+      error["cause"],
+    ]
+
+    for (const candidate of candidates) {
+      if (typeof candidate === "string" && candidate.length > 0) return candidate
+      if (
+        isRecord(candidate) &&
+        typeof candidate["message"] === "string" &&
+        candidate["message"].length > 0
+      ) {
+        return candidate["message"]
+      }
+    }
+  }
+
+  try {
+    return JSON.stringify(error)
+  } catch {
+    return String(error)
+  }
+}
+
+interface EventPropertiesLike {
+  [key: string]: unknown
+}
+
+export function getSessionErrorMessage(properties: EventPropertiesLike): string | undefined {
+  const errorRaw = properties["error"]
+  if (!isRecord(errorRaw)) return undefined
+
+  const dataRaw = errorRaw["data"]
+  if (isRecord(dataRaw)) {
+    const message = dataRaw["message"]
+    if (typeof message === "string") return message
+  }
+
+  const message = errorRaw["message"]
+  return typeof message === "string" ? message : undefined
+}