Dual US/Guatemalan national – traveling to Guatemala on Guatemalan passport, return on US Passport (will be mailed to traveler in Guate)

Here is my situation. My daughter is a US citizen born in Guatemala. My family is traveling in a couple weeks to Guatemala for the a service of my grandmother who dies of covid last November. She applied to renew her US passport back in April but they sent it back unprocessed a couple of weeks ago because she used the wrong form (she needed to use the DS-11 because her last passport was issued when she was 14). Anyway, she can’t get an appointment at a passport agency before then, She has tried multiple times- all booked up. She can try for an emergency appt three days before she leaves but not counting on that since it is not really an emergency. What about this idea? She redoes her US passport application expedited (6 weeks turnaround) in the US. She doesn’t wait for it but travels in 2 weeks to Guatemala on her Guatemalan passport with the rest of the family. Once her US passport in done (early August) I have it overnighted to her in Guatemala and she returns with it. Would that work? I know she is not supposed to leave the US without her US passport in hand but would anyone know? stop her? I thought about simply having her get a new passport issued to her at the US consulate in Guatemala but it looks like that would take a long time. She can’t exactly tell them her passport was stolen. She would have to admit she came down without it which probably mean they would expedite and appts seem hard to get there too. Thanks for any thoughts.