There are lots of gunsmiths that will shorten a stock for you. If she wants a 12 gauge, then you are going to have to budget for doing that.
Youth/Lady model guns all tend to be 20 gauge shotguns. If she is OK with shooting 3 inch mag 20 gauge loads for waterfowl, then you could get such a shotgun, and not have to make any changes to it.
The best new semiautomatic shotgun in your price range would be the Weatherby SA-08, which is quite a nice gun. Here is a link to an informational video: