My wife is 5'2", and a 15" would definitely be too big for her. Even with a mixte frame so she could step over the top tube(s) the reach is too far. I'd recommend a few visits to shops, and ask about kids' bikes. They'll be properly proportioned for smaller riders, although the components tend to be cheap and you might eventually want some of them to be swapped out. A bike with 507 (mountain) 24" wheels would probably do the trick.

Extra note: at that price range, DO NOT get suspension. A good shock will use up your entire budget by itself, which is another way of saying that an shock you get on a $300 bike will be cr@p.