Something to remember:
Kids are brutal on gear.
While my son has mostly very light gear, his bag is one area I just didn't worry about-I bought him one that can be used 9-10 months a year, and yes, synthetic. (I use down, but I am more careful!)
My son's pack, with water is 9-13 lbs loaded. So you can go light, even with heavier duty gear!
The North Face Tigger and the REI Zig Zag are both well made and affordable bags, in the 15-20* range. Kids tend to wander all over in the night, rolling off sleeping pads, scrunching down, etc. Quilts don't work so well, as kids come out of them, and can easily end up on cold ground.
The two bags I listed are not overly heavy as bags go, in the 2 lb to 2.4 lb range. Lightweight, maybe not-but they will last and have a good resale value.
Now, if you hike on hot areas, ignore what I said ;-) But if in the mountains, take a look!
PS: Also, don't forget that girls, like many ladies, sleep colder than men. So having a 20* bag may be a better choice than a 40* bag for her.
Some kids get into backpacking, others don't. Buying a $200-300 adult bag may end up being a pricey buy. My son does love backpacking, and has been on somewhere between 40-60 trips in his life…but, due to the fact that kids are stinky, I chose to go synthetic: I can wash it! I have told him he gets down when he hits teens (and crosses 5 ft tall, he is 4'7" now).