### 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_
actual_event = events[0]
        self.assert_valid_event(actual_event)
        assert actual_event["detail"] == EVENT_DETAIL
    def test_rule_disable(self, aws_client):
        rule_name = "rule-{}".format(short_uid())
        aws_client.events.put_rule(Name=rule_name, ScheduleExpression="rate(1 minutes)")
        response = aws_client.events.list_rules()
        assert response["Rules"][0]["State"] == "ENABLED"
        aws_client.events.disable_rule(Name=rule_name)
        response = aws_client.events.list_rules(NamePrefix=rule_name)
        assert response["Rules"][0]["State"] == "DISABLED"
        
        self.cleanup(rule_name=rule_name)
    def test_scheduled_expression_events(
        self,
        sns_create_topic,
        sqs_create_queue,
        sqs_queue_arn,
        sns_subscription,
        httpserver: HTTPServer,
        aws_client,
    ):
        httpserver.expect_request("").respond_with_data(b"", 200)
        http_endpoint = httpserver.url_for("/")
        topic_name = f"topic-{short_uid()}"
        queue_name = f"queue-{short_uid()}"
        fifo_queue_name = f"queue-{short_uid()}.fifo"
        rule_name = f"rule-{short_uid()}"
        sm_role_arn = arns.role_arn("sfn_role")
        sm_name = f"state-machine-{short_uid()}"