.NET: [Bug]: ChatClientAgent drops ResponseFormat before invoking underlying IChatClient

@moonbox3 is already working on this.

Since Sep 13, 2026.

### Description

### Minimal reproduction

The issue can be reproduced without relying on any provider-specific behavior.

A decorating `IChatClient` captures the `ChatOptions` that ultimately reach the underlying chat client.

When `ResponseFormat` is supplied through `AgentRunOptions`:

```csharp
var options = new AgentRunOptions
{
ResponseFormat =
ChatResponseFormat.ForJsonSchema()
};

await agent.RunAsync(
"Return a structured response.",
options: options);
```

the captured `ChatOptions.ResponseFormat` is `null`.

However, when the same response format is supplied directly to `IChatClient.GetResponseAsync`:

```csharp
var options = new ChatOptions
{
ResponseFormat =
ChatResponseFormat.ForJsonSchema()
};

await chatClient.GetResponseAsync(messages, options);
```

`ResponseFormat` reaches the underlying client correctly.

I originally discovered this using the Anthropic provider. Calling the Anthropic `IChatClient` directly correctly generates Anthropic's `output_config.format`, while routing the same request through `ChatClientAgent` / `AIAgent.RunAsync` causes the response format to disappear before the provider is invoked.

I also reproduced it:

* with and without tools
* on the initial model call
* after a tool-call/result turn
* using `RunAsync()`
* using explicit `AgentRunOptions.ResponseFormat`
* with the response format configured at agent construction time

This suggests the problem is in `ChatClientAgent` response-format propagation rather than in the Anthropic adapter or tool-calling path.

### Expected

`AgentRunOptions.ResponseFormat` and the response format implied by `RunAsync()` should be propagated into the `ChatOptions.ResponseFormat` passed to the underlying `IChatClient`.

### Actual

`ChatOptions.ResponseFormat` is absent when the underlying `IChatClient` is invoked.

### Questions

1. Is `ChatClientAgent` expected to propagate `AgentRunOptions.ResponseFormat` into the underlying `ChatOptions.ResponseFormat`?
2. Should `RunAsync()` result in the same provider-level structured-output configuration as directly setting `ChatOptions.ResponseFormat`?
3. Is there a supported workaround that preserves the `ChatClientAgent` pipeline while ensuring `ResponseFormat` reaches the underlying client?

### Code Sample

```markdown
using Microsoft.Agents.AI;
using Microsoft.Extensions.AI;

// The following is the error case
public sealed class CapturingChatClient : DelegatingChatClient
{
public ChatOptions? LastOptions { get; private set; }

public CapturingChatClient(IChatClient inner)
: base(inner)
{
}

public override Task GetResponseAsync(
IEnumerable messages,
ChatOptions? options = null,
CancellationToken cancellationToken = default)
{
LastOptions = options?.Clone();

return InnerClient.GetResponseAsync(
messages,
options,
cancellationToken);
}
}

public sealed record StructuredResponse(
string Message,
bool Success);

public static async Task ReproAsync(IChatClient rawClient)
{
var capturingClient = new CapturingChatClient(rawClient);

var agent = capturingClient.AsAIAgent(
name: "structured-output-repro",
instructions: "Return the requested structured response.");

var responseFormat =
ChatResponseFormat.ForJsonSchema();

var runOptions = new AgentRunOptions
{
ResponseFormat = responseFormat
};

await agent.RunAsync(
"Return Message='hello' and Success=true.",
options: runOptions);

Console.WriteLine(
capturingClient.LastOptions?.ResponseFormat is null
? "BUG: ResponseFormat was dropped"
: "OK: ResponseFormat reached IChatClient");
}

// The following is the working control case

public static async Task DirectChatClientControlAsync(
IChatClient chatClient)
{
var options = new ChatOptions
{
ResponseFormat =
ChatResponseFormat.ForJsonSchema()
};

await chatClient.GetResponseAsync(
[
new ChatMessage(
ChatRole.User,
"Return Message='hello' and Success=true.")
],
options);

// ResponseFormat reaches the underlying provider correctly here.
}
```

### Error Messages / Stack Traces

```markdown

```

### Package Versions

Microsoft.Agents.AI 1.19.0 and 1.21.0

### .NET Version

10.0.203

### Additional Context

_No response_