We had a really hard time finding smaller bikes for our kids that had gears, which was important for my son. They were more than I wanted to spend, about $250, but when they outgrow them you can trade them in for 100 off a new bike at the bike shop. Three years later we need to trade them in for larger bikes. Even though they are mountain bikes, I don't think they left the pavement often.
You could check at a bike shop and maybe get a nice used one.
Eizabeth