### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

IaC = CDK.

Attempting to connect an API-GW endpoint directly to EventBridge.

```
      const awsIntegration = new apigateway.AwsIntegration({
      service: 'events',
      action: 'PutEvents',
      integrationHttpMethod: 'POST',
      options: {
        credentialsRole: role,
        requestTemplates: {
          'application/json': requestTemplate,
        },
        passthroughBehavior: apigateway.PassthroughBehavior.NEVER,
        integrationResponses: [
          successResponse,
          serverErrorResponse,
          clientErrorResponse,
        ],
      },
    });
```

Receiving response:

```
    {
      statusCode: 400,
      data: {
        Type: 'User',
        message: `Error invoking integration for API Gateway ID '1z7ky5zzcb': API Gateway AWS integration action URI "arn:aws:apigateway:ap-southeast-2:events:action/PutEvents", method "GET" not yet implemented`,
        __type: 'InvalidRequest'
      }
    }
```

Similar to bug reported:

* https://github.com/localstack/localstack/issues/5508

### Expected Behavior

* 200 response
* Event created in relevant Eventbus

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

docker run localstack/localstack

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

Calling the API endpoint using axios from jest tests.

### Environment

```markdown
- OS: MacOS
- LocalStack: latest
```


### Anything else?

_No response_
keys = parsed_result.keys()
        if not ("statusCode" in keys and "body" in keys):
            LOG.warning(
                'Lambda output should follow the next JSON format: { "isBase64Encoded": true|false, "statusCode": httpStatusCode, "headers": { "headerName": "headerValue", ... },"body": "..."}'
            )
            response.status_code = 502
            response._content = json.dumps({"message": "Internal server error"})
            return response
        response.status_code = int(parsed_result.get("statusCode", 200))
        parsed_headers = parsed_result.get("headers", {})
        if parsed_headers is not None:
            response.headers.update(parsed_headers)
        try:
            result_body = parsed_result.get("body")
            if isinstance(result_body, dict):
                response._content = json.dumps(result_body)
            else:
                body_bytes = to_bytes(result_body or "")
                if parsed_result.get("isBase64Encoded", False):
                    body_bytes = base64.b64decode(body_bytes)
                response._content = body_bytes
        except Exception as e:
            LOG.warning("Couldn't set Lambda response content: %s", e)
            response._content = "{}"
        response.multi_value_headers = parsed_result.get("multiValueHeaders") or {}
        
        self.update_content_length(response)
        invocation_context.response = response
        self.response_templates.render(invocation_context)
        return invocation_context.response