It is NOT a difficult choice. It's a choice between two options that will results in the destruction of the option not chosen. A difficult choice is just called a difficult choice, IT HAS NO NAME because it doesn't deserve one for the dramatic effect.
A difficult choice is do you pick lover A or lover B and leave the other one possibly mad at you. A Sophie's choice is do you pick A or B and leave the other one DEAD, completely lifeless, future distinguished, no more fun experiences, and generally a bad existence (or lack thereof).