USB 3.0 disk not mounting in single user mode
Have you tried Safe Booting it? (Will bypass loading discreet graphics kext so you might be able to use GUI)
https://support.apple.com/en-au/HT201262

If you have another Mac with Thunderbolt handy you could put this one into Target Disk Mode and transfer your data through a Thunderbolt Cable?
https://support.apple.com/kb/ph10725

Otherwise it's likely the needed kext for USB3 isn't loaded in single user mode and you're out of luck and need a USB2 storage device.

As a side note the part needing to be replaced to resolve the graphics issue is the main logic board, separate from the storage, so I'm sure if you ask nicely to the Genius at the Apple Store to please not erase your data they won't.

See https://www.ifixit.com/Teardown/MacBook+Pro+15-Inch+Retina+Display+Late+2013+Teardown/18696 showing that it's a separate part.