refactor(prompt-callers): migrate team and call_omo_agent dispatch

Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)

Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>

src/features/team-mode/tools/messaging.ts

@@ -4,7 +4,7 @@ import { type ToolDefinition, tool } from "@opencode-ai/plugin/tool"
 import { z } from "zod"
 
 import type { TeamModeConfig } from "../../../config/schema/team-mode"
-import { promptAsyncAfterSessionIdle } from "../../../hooks/shared/prompt-async-gate"
+import { dispatchInternalPrompt } from "../../../hooks/shared/prompt-async-gate"
 import { log } from "../../../shared/logger"
 import { applyMemberSessionRouting, buildMemberPromptBody } from "../member-session-routing"
 import { buildEnvelope } from "../team-mailbox/poll"
@@ -199,7 +199,8 @@ async function deliverLive(
     applyMemberSessionRouting(recipientSessionId, recipientMember)
 
     try {
-      const promptResult = await promptAsyncAfterSessionIdle({
+      const promptResult = await dispatchInternalPrompt({
+        mode: "async",
         client,
         sessionID: recipientSessionId,
         source: "team-live-delivery",

src/tools/call-omo-agent/sync-executor.ts

@@ -1,6 +1,6 @@
 import type { PluginInput } from "@opencode-ai/plugin"
 import { clearSessionAgent, setSessionAgent, subagentSessions, syncSubagentSessions } from "../../features/claude-code-session-state"
-import { promptAsyncAfterSessionIdle } from "../../hooks/shared/prompt-async-gate"
+import { dispatchInternalPrompt } from "../../hooks/shared/prompt-async-gate"
 import { getAgentToolRestrictions, log } from "../../shared"
 import { getAgentDisplayName, stripAgentListSortPrefix } from "../../shared/agent-display-names"
 import {
@@ -134,7 +134,8 @@ export async function executeSync(
         return `Error: Failed to send prompt: promptAsync is not available on this OpenCode client.\n\n<task_metadata>\nsession_id: ${sessionID}\n</task_metadata>`
       }
 
-      const promptResult = await promptAsyncAfterSessionIdle({
+      const promptResult = await dispatchInternalPrompt({
+        mode: "async",
         client: ctx.client,
         sessionID,
         source: "call-omo-agent:sync",