### 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_
aws_client.events.put_rule(
            Name=rule_name,
            EventBusName=TEST_EVENT_BUS_NAME,
            EventPattern=json.dumps(pattern),
        )
        aws_client.events.put_targets(
            Rule=rule_name,
            EventBusName=TEST_EVENT_BUS_NAME,
            Targets=[{"Id": target_id, "Arn": queue_arn, "InputPath": "$.detail"}],
        )
        aws_client.events.put_events(Entries=[event])
        def get_message(queue_url):
            resp = aws_client.sqs.receive_message(QueueUrl=queue_url)
            return resp.get("Messages")
        messages = retry(get_message, retries=3, sleep=1, queue_url=queue_url)
        assert len(messages) == 1
        assert json.loads(messages[0].get("Body")) == json.loads(event["Detail"])
        event_details = json.loads(event["Detail"])
        event_details["admins"] = "no"
        event["Detail"] = json.dumps(event_details)
        aws_client.events.put_events(Entries=[event])
        messages = retry(get_message, retries=3, sleep=1, queue_url=queue_url)
        assert messages is None
        
        self.cleanup(TEST_EVENT_BUS_NAME, rule_name, target_id, queue_url=queue_url)
    @pytest.mark.parametrize(
        "schedule_expression", ["rate(1 minute)", "rate(1 day)", "rate(1 hour)"]
    )
    @pytest.mark.aws_validated
    def test_create_rule_with_one_unit_in_singular_should_succeed(