### 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,
        aws_client,
        sqs_create_queue,
        sqs_queue_arn,
        events_put_rule,
        events_allow_event_rule_to_sqs_queue,
        snapshot,
    ):
        default_bus_rule_name = f"rule-{short_uid()}"
        default_bus_target_id = f"test-target-default-b-{short_uid()}"
        nonexistent_event_bus = f"event-bus-{short_uid()}"
        snapshot.add_transformer(
            [
                snapshot.transform.key_value("MD5OfBody"),  
                snapshot.transform.key_value("ReceiptHandle"),
                snapshot.transform.regex(nonexistent_event_bus, "<custom-event-bus>"),
            ]
        )
        
        
        queue_url = sqs_create_queue()
        queue_arn = sqs_queue_arn(queue_url)
        rule_on_default_bus = events_put_rule(
            Name=default_bus_rule_name,