I think Yuri is going to die too... every episode makes it see more and more likely. However, if somehow the second ending does happen, documents and identification might not be a huge problem.
I don't know if it's the same in Korea but in America there are plenty of cases where people live under new name by stealing someone's identity. They find someone who has died, usually they pick a baby or toddler who had been born in a year close to their own and request for the child's birth certificate. Once that is on hand, they can request other forms of identification. Yuri doesn't need to do all.
Plus, If that doesn't work out, she can probably prove to the government through other methods that she is Yuri. If she submits enough proof, I doubt the government will question why she is not dead. They could assume it was a glitch in the system which classified her as 'dead' due to some sort of error.
I'm sure MinJung would feel betrayed but I have faith they would be able to work it out. MinJung has already been proven to be very nice... given sometime there's an enormous possibility she will understand why she was not told the truth. The bigger problem would be how they would explain this to everyone who attended MC's funeral.
