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/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,