refactor(background-agent): use unified internal prompt dispatch

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

@@ -6,7 +6,7 @@ import {
   clearAllDelegatedChildSessionBootstrap,
   getDelegatedChildSessionBootstrap,
 } from "../../shared/delegated-child-session-bootstrap"
-import { promptAsyncAfterSessionIdle } from "../../shared/prompt-async-gate"
+import { dispatchInternalPrompt } from "../../shared/prompt-async-gate"
 import { clearSessionPromptParams, getSessionPromptParams } from "../../shared/session-prompt-params-state"
 import {
   getSessionAgent,
@@ -2516,7 +2516,8 @@ describe("BackgroundManager.resume promptAsync gate state", () => {
         abort: async () => ({}),
       },
     }
-    await promptAsyncAfterSessionIdle({
+    await dispatchInternalPrompt({
+      mode: "async",
       client,
       sessionID: "session-reserved-resume",
       source: "test-existing-reservation",

src/features/background-agent/manager.ts

@@ -3,7 +3,7 @@ import type { PluginInput } from "@opencode-ai/plugin"
 import type { BackgroundTaskConfig, TmuxConfig } from "../../config/schema"
 import { setContinuationMarkerSource } from "../../features/run-continuation-state"
 import type { ModelFallbackControllerAccessor } from "../../hooks/model-fallback"
-import { type PromptAsyncGateResult, promptAsyncAfterSessionIdle } from "../../hooks/shared/prompt-async-gate"
+import { dispatchInternalPrompt, type PromptAsyncGateResult } from "../../hooks/shared/prompt-async-gate"
 import { isSessionActive as isOpenCodeSessionActive } from "../../hooks/shared/session-idle-settle"
 import {
   createInternalAgentTextPart,
@@ -1290,7 +1290,8 @@ The fallback retry session is now created and can be inspected directly.
       applySessionPromptParams(existingTask.sessionId!, existingTask.model)
     }
 
-    promptAsyncAfterSessionIdle({
+    dispatchInternalPrompt({
+      mode: "async",
       client: this.client,
       sessionID: existingTask.sessionId,
       source: "background-agent-resume",

src/features/background-agent/parent-wake-notifier.ts

@@ -1,7 +1,7 @@
 import { resolveRegisteredAgentName } from "../claude-code-session-state"
 import { createInternalAgentTextPart, log, messagesInDirectory, normalizeSDKResponse } from "../../shared"
 import { isSessionActive as isOpenCodeSessionActive, settleAfterSessionIdle } from "../../hooks/shared/session-idle-settle"
-import { promptAsyncAfterSessionIdle } from "../../hooks/shared/prompt-async-gate"
+import { dispatchInternalPrompt } from "../../hooks/shared/prompt-async-gate"
 import type { PluginInput } from "@opencode-ai/plugin"
 
 type OpencodeClient = PluginInput["client"]
@@ -137,7 +137,8 @@ export class ParentWakeNotifier {
     const notificationContent = latestWake.notifications.join("\n\n")
 
     try {
-      const promptResult = await promptAsyncAfterSessionIdle({
+      const promptResult = await dispatchInternalPrompt({
+        mode: "async",
         client: this.deps.client,
         sessionID,
         source: "background-agent-parent-wake",

src/features/background-agent/session-route.ts

@@ -1,7 +1,7 @@
 import type { PluginInput } from "@opencode-ai/plugin"
 
 import { promptWithModelSuggestionRetry } from "../../shared"
-import { promptAsyncAfterSessionIdle } from "../../shared/prompt-async-gate"
+import { dispatchInternalPrompt } from "../../shared/prompt-async-gate"
 
 type OpencodeClient = PluginInput["client"]
 
@@ -35,7 +35,8 @@ export function promptAsyncInDirectory(
     return Promise.reject(new Error("session id is required for routed promptAsync"))
   }
 
-  return promptAsyncAfterSessionIdle({
+  return dispatchInternalPrompt({
+    mode: "async",
     client,
     sessionID,
     input: routedArgs,