feat: add public get_mcp_status() method to ClaudeSDKClient (#516)

## Motivation

The Claude Investigator app, which uses the python claude agent SDK,
needs to query MCP server connection status (example
[here](https://github.com/anthropics/anthropic/blob/948dc8cc4a739b7d026d54e8da2e175cd1659340/ts-threat-intel/ts_threat_intel/claude_investigator/services/client_actor.py#L554-L557)).
Currently it does this by reaching into private SDK internals:

```python
if client._query and hasattr(client._query, "_send_control_request"):
    mcp_status_response = await client._query._send_control_request(
        {"subtype": "mcp_status"}
    )
```

This is fragile — any SDK refactor silently breaks the caller, and the
`hasattr` guard means failures degrade silently to no MCP status
reporting.

## Changes

Add a public `get_mcp_status()` method following the exact same pattern
as `interrupt()`, `set_permission_mode()`, `set_model()`, and
`rewind_files()`:

- **`Query.get_mcp_status()`** — sends `{"subtype": "mcp_status"}`
control request via `_send_control_request`
- **`ClaudeSDKClient.get_mcp_status()`** — public wrapper with
connection check, docstring with Returns and Example sections

Callers can now simply do:

```python
mcp_status_response = await client.get_mcp_status()
```

## Testing

The method delegates entirely to the existing `_send_control_request`
infrastructure which is already tested. The new code is a 3-line wrapper
with no branching logic.

Author: mdworsky

src/claude_agent_sdk/_internal/query.py

@@ -517,6 +517,10 @@ async def _handle_sdk_mcp_request(
                 "error": {"code": -32603, "message": str(e)},
             }
 
+    async def get_mcp_status(self) -> dict[str, Any]:
+        """Get current MCP server connection status."""
+        return await self._send_control_request({"subtype": "mcp_status"})
+
     async def interrupt(self) -> None:
         """Send interrupt control request."""
         await self._send_control_request({"subtype": "interrupt"})

src/claude_agent_sdk/client.py

@@ -293,6 +293,32 @@ async def rewind_files(self, user_message_id: str) -> None:
             raise CLIConnectionError("Not connected. Call connect() first.")
         await self._query.rewind_files(user_message_id)
 
+    async def get_mcp_status(self) -> dict[str, Any]:
+        """Get current MCP server connection status (only works with streaming mode).
+
+        Queries the Claude Code CLI for the live connection status of all
+        configured MCP servers.
+
+        Returns:
+            Dictionary with MCP server status information. Contains a
+            'mcpServers' key with a list of server status objects, each having:
+            - 'name': Server name (str)
+            - 'status': Connection status ('connected', 'pending', 'failed',
+              'needs-auth', 'disabled')
+
+        Example:
+            ```python
+            async with ClaudeSDKClient(options) as client:
+                status = await client.get_mcp_status()
+                for server in status.get("mcpServers", []):
+                    print(f"{server['name']}: {server['status']}")
+            ```
+        """
+        if not self._query:
+            raise CLIConnectionError("Not connected. Call connect() first.")
+        result: dict[str, Any] = await self._query.get_mcp_status()
+        return result
+
     async def get_server_info(self) -> dict[str, Any] | None:
         """Get server initialization info including available commands and output styles.