I no longer get the option to back up an external drive on my Mac

Hey Spillage, I'd like for us to try to enable the system warnings on your computer, in case these have been disabled. This could be the issue with the prompt that doesn't appear at all. So, I will need you to click on the small Dropbox icon on the taskbar.
 

And then, please click on your profile picture/initials at the top right of the window that appears. Next, click on Preferences, and from the tabs at the top, please select the ''Account'' tab.
Once you are in the Account tab, I need you to press Alt Key + H on your keyboard.

Once you "reset file system warning options", please unplug and plug the external drive again, to see if the prompt appears.

Also, to answer your previous questions: you'd need to backup the external hard drive, not just one, single folder. 

Also, when you say you wish to keep the external drive's files where they are, would you let me know a bit more? Do you mean inside the drive, or synced locally on the device?