Hi there!
Do you have access to the notification sound file? If so, there is a notifications file that you can place “custom” sounds, and once they are in that file they should appear in your sounds listing. Once you get the sound onto your Galaxy, go to the “My Files” app in your Samsung app collection, and select “Internal Storage”. Scroll down, and you will see a “Notifications” folder. You need to move the sound file from where it is currently located (in Downloads probably) into this folder.
Once this is done, go back to your Ring app, and you should find the sound available to use.
Hope this helps!