### 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_
def _allow_event_rule(sqs_queue_url, sqs_queue_arn, event_rule_arn) -> None:
        
        aws_client.sqs.set_queue_attributes(
            QueueUrl=sqs_queue_url,
            Attributes={
                "Policy": json.dumps(
                    {
                        "Statement": [
                            {
                                "Sid": "AllowEventsToQueue",
                                "Effect": "Allow",
                                "Principal": {"Service": "events.amazonaws.com"},
                                "Action": "sqs:SendMessage",
                                "Resource": sqs_queue_arn,
                                "Condition": {"ArnEquals": {"aws:SourceArn": event_rule_arn}},
                            }
                        ]
                    }
                )
            },
        )
    return _allow_event_rule
@pytest.fixture
def events_put_rule(aws_client):
    rules = []
    def _factory(**kwargs):