You are right that a CX style of bike would work well, but at 5'3", your GF needs a small wheeled machine. There is a limit to how small you can make a 700C frame without bodging the geometry.
26" MTB wheels in a lightweight frame are ideal: check out some of the Thorn bikes at sjscycles.com for inspiration but they start at twice your budget. Lightweight 26" wheelers are niche products, but you may be able to get a used one in the back pages of the CTC magazine or from SJS cycles.

A mid-range Al-frames MTB with front sus will probably do the job. MTB frames are not so heavy these days.Specialized Hardrock seems to be the base level for serious riding.