[Bug]: Frontend tools registered via useFrontendTool are not invokable when using CopilotKit ↔ A2A (A2AAgent + A2AClient), but work via HttpAgent + ag_ui_adk endpoint

### Pre-flight Checklist

- [x] I have searched [existing issues](https://github.com/ag-ui-protocol/ag-ui/issues) and this hasn't been reported yet.
- [x] I am using the **latest** version AG-UI.

### Describe the Bug

When I register a client-side tool in CopilotKit using `useFrontendTool` (example tool: `sayHello`) and expose tools on the backend via `AGUIToolset()` in a Google ADK A2A agent, the agent is **unable to invoke** the frontend tool when communicating through:

* CopilotKit runtime (`@copilotkit/runtime/v2`)
* `A2AAgent` (`@ag-ui/a2a`)
* `A2AClient` TypeScript client (`@a2a-js/sdk/client`)

However, if I expose the backend ADK agent using `ag_ui_adk` (`add_adk_fastapi_endpoint`) and connect from CopilotKit using `HttpAgent` (`@ag-ui/client`), the same frontend tool **is invoked successfully**.

This suggests tool invocation routing/handshake is broken specifically in the **A2AAgent + A2AClient path**, not in CopilotKit frontend tool registration itself.

---

### Affected Components

* CopilotKit frontend tool registration: `useFrontendTool`
* CopilotKit runtime endpoint: `createCopilotEndpoint` + `CopilotRuntime`
* A2A integration: `A2AAgent` + `A2AClient`
* Backend: Google ADK A2A agent with `AGUIToolset()`

### Actual behavior

When using **A2AAgent + A2AClient** as the runtime agent backend, the agent does **not** invoke the frontend tool (tool call does not reach the browser handler).

When using **HttpAgent + add_adk_fastapi_endpoint**, the tool invocation works and the browser tool handler executes.

## Why this seems like a bug

* The frontend tool is registered correctly (works in the HttpAgent path).
* The backend exposes tool support (`AGUIToolset()`).
* The only difference is transport/protocol path:

* ✅ Works: CopilotKit ↔ HttpAgent ↔ `ag_ui_adk` endpoint
* ❌ Fails: CopilotKit ↔ `A2AAgent` ↔ `A2AClient` ↔ A2A server

This suggests the A2A integration is not forwarding/negotiating client tool definitions or not routing tool-call messages back to the CopilotKit frontend tool executor.

---

### Steps to Reproduce

---

### 1) Register a frontend tool (Next.js client component)

`page.tsx`:

```tsx
"use client";

import { CopilotChat, CopilotKitProvider } from "@copilotkit/react-core/v2";
import { useFrontendTool } from "@copilotkit/react-core";
import { createA2UIMessageRenderer } from "@copilotkit/a2ui-renderer";
import { theme } from "./theme";

export const dynamic = "force-dynamic";

function MCPTool() {
useFrontendTool({
name: "sayHello",
description: "Say hello to someone.",
parameters: [
{
name: "name",
type: "string",
description: "name of the person to greet",
required: true,
},
],
handler: async ({ name }) => {
alert(`Hello, ${name}!`);
},
});

return null;
}

export default function Home() {
const A2UIMessageRenderer = createA2UIMessageRenderer({ theme });
const activityRenderers = [A2UIMessageRenderer];

return (




);
}
```

### 2) Create a CopilotKit runtime endpoint using A2AAgent + A2AClient

`api/copilotkit/route.ts`:

```ts
import {
CopilotRuntime,
createCopilotEndpoint,
InMemoryAgentRunner,
} from "@copilotkit/runtime/v2";
import { handle } from "hono/vercel";
import { A2AAgent } from "@ag-ui/a2a";
import { A2AClient } from "@a2a-js/sdk/client";

export const GET = (req: Request) => handler(req);
export const POST = (req: Request) => handler(req);

async function handler(req: Request) {
const a2aClient = new A2AClient("http://localhost:10007");
const agent = new A2AAgent({ a2aClient });

const runtime = new CopilotRuntime({
agents: { my_agent: agent },
runner: new InMemoryAgentRunner(),
});

const app = createCopilotEndpoint({
runtime,
basePath: "/api/copilotkit",
});

return handle(app)(req);
}
```

### 3) Backend ADK A2A agent includes `AGUIToolset()`

```py
def _build_agent(self) -> LlmAgent:
root_agent = LlmAgent(
name="basic_a2a_agent",
description="Helpful assistant",
model=LiteLlm(
model=f"openai/{os.getenv('OPENAI_MODEL_NAME')}",
api_key=os.getenv("OPENAI_API_KEY")
),
instruction="You are helpful agent, answer to user query in polite and friendly manner",
tools=[AGUIToolset()],
)
adk_agent = ADKAgent(
adk_agent=root_agent,
app_name="agents",
user_id="demo_user",
)
return adk_agent
```

### 4) Test: ask the agent to call `sayHello`

Example prompt:

> “Call the sayHello tool with name=Suraj”

**Result:** No alert appears, tool handler not invoked.

---

## Control test (works)

If I expose the backend agent with:

```py
add_adk_fastapi_endpoint(app, adk_agent, path="/")
```

…and connect CopilotKit using `HttpAgent` (AG-UI protocol), then the frontend tool invocation works as expected.

---

### Expected Behavior

---

When the user asks the agent to call a frontend tool (e.g., “Say hello to Suraj”), the backend agent should be able to invoke the CopilotKit client tool registered via `useFrontendTool`, and the handler should run in the browser:

```ts
handler: async ({ name }) => alert(`Hello, ${name}!`)
```

---

### Environment

```text
---

* Backend: "a2a-sdk",
"google-adk",
"google-genai",
"python-dotenv",
"litellm",
"ag-ui-adk>=0.4.2",

* Package versions:

"@a2a-js/sdk": "0.2.5",
"@a2ui/lit": "^0.8.1",
"@ag-ui/a2a": "0.00.6",
"@copilotkit/a2ui-renderer": "^1.51.3",
"@copilotkit/react-core": "^1.51.3",
"@copilotkit/react-ui": "^1.51.3",
"@copilotkit/runtime": "^1.51.3",
"hono": "^4.6.18",
"next": "^16.0.10",
"react": "^19.2.3",
"react-dom": "^19.2.3",
"zod": "^3.25.75"

---
```

### Screenshots

_No response_

### Logs & Errors

```shell
For A2A ADK agent

-2026-02-18 13:06:44,972 - INFO - ag_ui_adk.client_proxy_toolset - Initialized ClientProxyToolset with 0 tools (all long-running)
- 2026-02-18 13:06:44,972 - DEBUG - ag_ui_adk.adk_agent - Replaced AGUIToolset with ClientProxyToolset for agent basic_a2a_agent

for ag_ui_adk agent

- 2026-02-18 13:09:31,239 - INFO - ag_ui_adk.client_proxy_toolset - Initialized ClientProxyToolset with 1 tools (all long-running)
- 2026-02-18 13:09:31,239 - DEBUG - ag_ui_adk.adk_agent - Replaced AGUIToolset with ClientProxyToolset for agent basic_a2a_agent
```

### Additional Context

_No response_