Yeah, I'd be inclined to spend a couple hundred more and get one of those ultra-cheap 50cc Chinese scooters you see online with free shipping for $700-800, or at least a newer used scooter.
I mean, I know scooters are selling for a lot right now with the high gas prices and all, but in the last week locally there was a 10 year old CH80 (bigger than you want, I know) for $800 and a 2003 Metropolitan for $900, for example, plus some no-name mopeds. And this is in a small town at the ass end of nowhere -- if you have a bigger metro area nearby, you have more options.
So I'd look for the newest and cleanest 50cc scooter you can get for around your price range, plus a couple of hundred (since an old bike will need some work, pretty much guaranteed).
I don't know what defines a "moped" versus a "motorcycle" where you live -- in most places, it's a "moped" if it has less than 50cc's and a top speed of less than about 35, but those definitions can get pretty slippery. But anyway, what I'm saying is that it doesn't necessarily need to have pedals to be a "moped," and getting her on a more modern scooter with decent brakes and lights and more reliability would be good for your peace of mind.
Edited to add: I just took a look at the photo on the wikipedia page, and there's a guy here in town who rides one of those. It's SLOW, even compared to other small scooters, when he takes off from a traffic light. Definitely ride it before committing yourself -- if they are all that slow, you don't want her out on a fast rural road on one of those.