You can always export the service to filegdb. This would preserve GlobalIds, rel, and attachments.
Then you can use the filegdb to publish a new service with a new name.
Using Append API will allow to transfer the data from the filegdb to the new service without creating new service. It will preserve the GlobalId also but it will not create any relationships or attachments. So importing the filegdb to a new service is the best option.
Khaled Hassen
Online Feature Service Dev Lead