### 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_
)
        except ClientError as e:
            raise IntegrationAccessError() from e
        response = LambdaResponse()
        if asynchronous:
            response._content = ""
        else:
            
            match result:
                case str():
                    
                    
                    if isinstance(json.loads(result or "{}"), dict):
                        parsed_result = result
                    else:
                        
                        
                        parsed_result = json.loads(result or "{}")
                case _:
                    parsed_result = json.loads(str(result or "{}"))
            parsed_result = common.json_safe(parsed_result)
            parsed_result = {} if parsed_result is None else parsed_result
            response._content = parsed_result
        response.status_code = 200
        
        invocation_context.response = response
        self.response_templates.render(invocation_context)
        invocation_context.response.headers["Content-Length"] = str(len(response.content or ""))
        return invocation_context.response
class KinesisIntegration(BackendIntegration):