### 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_
events_client = aws_stack.create_external_boto_client("events", region_name="eu-west-1")
        response = events_client.put_events(Entries=[{"DetailType": "Test", "Detail": "{}"}])
        assert response.get("Entries")
    def test_put_event_without_detail(self):
        events_client = aws_stack.create_external_boto_client("events", region_name="eu-west-1")
        response = events_client.put_events(
            Entries=[
                {
                    "DetailType": "Test",
                }
            ]
        )
        assert response.get("Entries")
    def test_trigger_event_on_ssm_change(self, aws_client):
        rule_name = "rule-{}".format(short_uid())
        target_id = "target-{}".format(short_uid())
        
        queue_name = "queue-{}".format(short_uid())
        queue_url = aws_client.sqs.create_queue(QueueName=queue_name)["QueueUrl"]
        queue_arn = arns.sqs_queue_arn(queue_name)
        
        ssm_prefix = "/test/local/"
        aws_client.events.put_rule(
            Name=rule_name,