### 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_
self,
        bus_name=None,
        rule_name=None,
        target_ids=None,
        queue_url=None,
        events_client=None,
        sqs_client=None,
        log_group_name=None,
        logs_client=None,
    ):
        events_client = events_client or aws_stack.create_external_boto_client("events")
        kwargs = {"EventBusName": bus_name} if bus_name else {}
        if target_ids:
            target_ids = target_ids if isinstance(target_ids, list) else [target_ids]
            events_client.remove_targets(Rule=rule_name, Ids=target_ids, Force=True, **kwargs)
        if rule_name:
            events_client.delete_rule(Name=rule_name, Force=True, **kwargs)
        if bus_name:
            events_client.delete_event_bus(Name=bus_name)
        if queue_url:
            sqs_client = sqs_client or aws_stack.create_external_boto_client("sqs")
            sqs_client.delete_queue(QueueUrl=queue_url)
        if log_group_name:
            logs_client = logs_client or aws_stack.create_external_boto_client("logs")
            log_streams = logs_client.describe_log_streams(logGroupName=log_group_name)
            for log_stream in log_streams["logStreams"]:
                logs_client.delete_log_stream(