### 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_
EventPattern=json.dumps({"detail-type": ["CustomType"], "source": ["MySource"]}),
            State="ENABLED",
        )
        events_allow_event_rule_to_sqs_queue(
            event_rule_arn=rule_on_default_bus["RuleArn"],
            sqs_queue_arn=queue_arn,
            sqs_queue_url=queue_url,
        )
        aws_client.events.put_targets(
            Rule=default_bus_rule_name,
            Targets=[{"Id": default_bus_target_id, "Arn": queue_arn}],
        )
        
        
        entries = [
            {
                "Source": "MySource",
                "DetailType": "CustomType",
                "Detail": json.dumps({"message": "for the default event bus"}),
            },
            {
                "EventBusName": nonexistent_event_bus,
                "Source": "MySource",
                "DetailType": "CustomType",
                "Detail": json.dumps({"message": "for the custom event bus"}),
            },
        ]
        response = aws_client.events.put_events(Entries=entries)
        snapshot.match("put-events", response)
        def _get_sqs_messages():