She'll still get a bike that fits her better if she can test ride it first.

Perhaps your GF is one of those people who need to go through a 'bargain bike' phase the first time around they get a bike. It's pretty common. Then as they ride more, they slowly get to experience the little things that might be lacking (appropriate frame, geometry, weight, fit, good quality components...). If the biking catches on with her, she'll outgrow the cheapo bike in about a year and want something that feels and works smoother for her. That's not the end of the world.
Sometimes you just have to let people find out stuff in their own time.

There ARE plenty of $350 new bikes around in bike shops- I still feel it's more important to have her test ride her new bike than it is to take a chance on a better used bike bought without a test ride. There are so many variables besides geometry. Others might feel differently.