fix(background-agent): keep delegated skill, permission, and child agent across retries

Three coupled gaps surfaced after the initial spawn fix:

1. fallback-retry-handler dropped task.skillContent and
   task.sessionPermission when rebuilding LaunchInput, so the retried
   background task lost the delegated system prompt and question-deny
   permission rule.
2. manager.startTask never bound the child sessionID to the resolved
   agent via setSessionAgent, leaving runtime fallback and other hooks
   with no idea which agent owned the new child session.
3. The fallback-to-general path in spawner.ts rebuilt the prompt body
   without going through buildFallbackBody, so bootstrap state, session
   tools, and session agent updates drifted apart.

Persist skillContent and sessionPermission on BackgroundTask, bind
setSessionAgent/updateSessionAgent at session creation and on fallback,
and route the FALLBACK_AGENT retry through buildFallbackBody so the
prompt body, bootstrap tools, and session registries all agree.

src/features/background-agent/manager.ts

@@ -28,7 +28,7 @@ import { SessionCategoryRegistry } from "../../shared/session-category-registry"
 import { applySessionPromptParams } from "../../shared/session-prompt-params-helpers"
 import { setSessionTools } from "../../shared/session-tools-store"
 import { isInsideTmux } from "../../shared/tmux"
-import { subagentSessions } from "../claude-code-session-state"
+import { setSessionAgent, subagentSessions, updateSessionAgent } from "../claude-code-session-state"
 import { MESSAGE_STORAGE } from "../hook-message-injector"
 import { getTaskToastManager } from "../task-toast-manager"
 import { abortWithTimeout } from "./abort-with-timeout"
@@ -574,6 +574,8 @@ export class BackgroundManager {
         parentTools: input.parentTools,
         model: input.model,
         fallbackChain: input.fallbackChain,
+        skillContent: input.skillContent,
+        sessionPermission: input.sessionPermission,
         attemptCount: 0,
         category: input.category,
         onSessionCreated: input.onSessionCreated,
@@ -756,6 +758,7 @@ export class BackgroundManager {
     await input.onSessionCreated?.(sessionID)
     this.settlePreStartDescendantReservation(task)
     subagentSessions.add(sessionID)
+    setSessionAgent(sessionID, input.agent)
 
     if (this.tasks.get(task.id)?.status === "cancelled") {
       clearDelegatedChildSessionBootstrap(sessionID)
@@ -826,12 +829,39 @@ The fallback retry session is now created and can be inspected directly.
     this.taskHistory.record(input.parentSessionId, { id: task.id, sessionID, agent: input.agent, description: input.description, status: "running", category: input.category, startedAt: task.startedAt })
     this.startPolling()
 
+    // Fire-and-forget prompt via promptAsync (no response body needed)
+    // OpenCode prompt payload accepts model provider/model IDs and top-level variant only.
+    // Temperature/topP and provider-specific options are applied through chat.params.
+    const launchModel = input.model
+      ? {
+          providerID: input.model.providerID,
+          modelID: input.model.modelID,
+        }
+      : undefined
+    const launchVariant = input.model?.variant
+
+    if (input.model) {
+      applySessionPromptParams(sessionID, input.model)
+    }
+
+    const launchTools = {
+      task: false,
+      call_omo_agent: true,
+      question: false,
+      ...getAgentToolRestrictions(input.agent, {
+        includeTeamToolDenylist: input.teamRunId === undefined,
+      }),
+    }
+    setSessionTools(sessionID, launchTools)
+
     log("[background-agent] Launching task:", { taskId: task.id, sessionID, agent: input.agent })
     registerDelegatedChildSessionBootstrap({
       sessionID,
       promptText: input.prompt,
       fallbackChain: input.fallbackChain,
       category: input.category,
+      system: input.skillContent,
+      tools: launchTools,
       modelFallbackControllerAccessor: this.modelFallbackControllerAccessor,
     })
 
@@ -848,38 +878,12 @@ The fallback retry session is now created and can be inspected directly.
       promptLength: input.prompt.length,
     })
 
-    // Fire-and-forget prompt via promptAsync (no response body needed)
-    // OpenCode prompt payload accepts model provider/model IDs and top-level variant only.
-    // Temperature/topP and provider-specific options are applied through chat.params.
-    const launchModel = input.model
-      ? {
-          providerID: input.model.providerID,
-          modelID: input.model.modelID,
-        }
-      : undefined
-    const launchVariant = input.model?.variant
-
-    if (input.model) {
-      applySessionPromptParams(sessionID, input.model)
-    }
-
     const promptBody = {
       agent: input.agent,
       ...(launchModel ? { model: launchModel } : {}),
       ...(launchVariant ? { variant: launchVariant } : {}),
       system: input.skillContent,
-      tools: (() => {
-        const tools = {
-          task: false,
-          call_omo_agent: true,
-          question: false,
-          ...getAgentToolRestrictions(input.agent, {
-            includeTeamToolDenylist: input.teamRunId === undefined,
-          }),
-        }
-        setSessionTools(sessionID, tools)
-        return tools
-      })(),
+      tools: launchTools,
       parts: [createInternalAgentTextPart(input.prompt)],
     }
 
@@ -898,7 +902,18 @@ The fallback retry session is now created and can be inspected directly.
           const fallbackBody = buildFallbackBody(promptBody, FALLBACK_AGENT, {
             includeTeamToolDenylist: input.teamRunId === undefined,
           })
-          setSessionTools(sessionID, fallbackBody.tools as Record<string, boolean>)
+          const fallbackTools = fallbackBody.tools as Record<string, boolean>
+          setSessionTools(sessionID, fallbackTools)
+          updateSessionAgent(sessionID, FALLBACK_AGENT)
+          registerDelegatedChildSessionBootstrap({
+            sessionID,
+            promptText: input.prompt,
+            fallbackChain: input.fallbackChain,
+            category: input.category,
+            system: input.skillContent,
+            tools: fallbackTools,
+            modelFallbackControllerAccessor: this.modelFallbackControllerAccessor,
+          })
           await promptWithRetryInDirectory(this.client, {
             path: { id: sessionID },
             body: fallbackBody,