I don't have any experience with an underbite per se, but I have had very extensive orthodontic and orthognathic work done on my own jaws to correct an unexplained growth spurt that occurred on one side of my jaw when I got my braces off at the age of 14. There is no explanation for why my lower jaw grew as it did as both of my parents have naturally nice straight teeth, and there is no other orthognathic history that we know of that could have contributed to it. My upper jaw became slightly tilted in order to accommodate my lower jaw's growth. My mouth was truly a mess.
I wore a headgear when I had braces as a pre-teen, and it was limited to night wearings only. I had to wear a strap that went around the crown of my head and another strap that went around the nape of my neck. I don't have any bad memories of being teased about the headgear and braces and got more grief for my glasses than anything else I can recall.
You will most definitely want to have her underbite corrected as it can lead to a long history of dental and periodontic issues. I've had braces again and two upper and lower jaw surgeries and am happy to have the whole ordeal behind me although the structure of my teeth themselves has been weakened due to the 15 years I lived with uncorrected jaws.