I got my large 8 year old 70 lbs on a small 26 inch Dawes Haymaker 1000 for 300 bucks. It has surprisingly reasonable components for the price, is easy to ride and he had no problem with the derailleur shifting. You may be able to find one used in your target price. Bikesdirect.com sells them. I put smaller street tires on it to make it a great townie bike, which also lowered it a bit too. Not what you were looking for, but propaply a great solution for her.
http://www.bikesdirect.com/products/dawes/hay1000xi.htm.
X small size
It has a rock solid Shimano 7 speed rear and triple front.. He is now twelve and tours with us on the same bike. He wiki be approaching a thousand miles on it and it has had zero maintenance since it came out of the box four years ago.