Something similar happened to me while in Europe. I had the GC with me, and crossed the security check in. Then, I lost the GC inside the airport. So, when I was getting ready to board the plane, I realized, I do not have the GC anymore. I informed the airline employee about what happened. Apparently, in Europe, the US government people are in the airport always, so a guy representing the Federal government of the US came to the gate within 15 minutes. He asked for my A number. To verify my identity, he asked his coworker in Washington to text him the picture and details of that A number. He got a text with my picture and details, which matched with my state ID. So, he asked the airline employee to let me board the plane, and I did not miss the flight.

May be, something similar happens in Nepal too. I am guessing the whole embassy can't take a day off without an emergency-handler. So, you might want to ask your friend to get in touch with the airlines or look for emergency US embassy line.