### 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_
finally:
            self.cleanup(
                bus_name,
                rule_name,
                target_id,
                queue_url=queue_url,
                events_client=events_client,
                sqs_client=sqs_client,
            )
    def _put_entries_assert_results_sqs(
        self, events_client, sqs_client, queue_url: str, entries: List[Dict], should_match: bool
    ):
        response = events_client.put_events(Entries=entries)
        assert not response.get("FailedEntryCount")
        def get_message(queue_url):
            resp = sqs_client.receive_message(QueueUrl=queue_url)
            messages = resp.get("Messages")
            if should_match:
                assert len(messages) == 1
            return messages
        messages = retry(get_message, retries=5, sleep=1, queue_url=queue_url)
        if should_match:
            actual_event = json.loads(messages[0]["Body"])
            if "detail" in actual_event:
                self.assert_valid_event(actual_event)
        else:
            assert not messages
        return messages
    
    def test_put_events_with_target_sns(self, sns_subscription, aws_client):