So what was the diaper lady reference?

It's just a reference to Carmen's bedwetting. It's quite common in children and it can last until the child even turns the age Carmen was in the movie (in my case, it ended when I was 7, so I understood the reference quite well). The only way to prevent the child's bed from getting wet is to have them wear diapers at night, which is why Juni called Carmen a "diaper lady". Hope this helped!