### 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_
return os.path.join(config.dirs.tmp, EVENTS_TMP_DIR)
def _create_and_register_temp_dir():
    tmp_dir = _get_events_tmp_dir()
    if not os.path.exists(tmp_dir):
        mkdir(tmp_dir)
        TMP_FILES.append(tmp_dir)
    return tmp_dir
def _dump_events_to_files(events_with_added_uuid):
    try:
        _create_and_register_temp_dir()
        current_time_millis = int(round(time.time() * 1000))
        for event in events_with_added_uuid:
            target = os.path.join(
                _get_events_tmp_dir(),
                "%s_%s" % (current_time_millis, event["uuid"]),
            )
            save_file(target, json.dumps(event["event"]))
    except Exception as e:
        LOG.info("Unable to dump events to tmp dir %s: %s", _get_events_tmp_dir(), e)
def handle_numeric_conditions(conditions: List[Any], value: float):