You'll have a hard time finding a reliable self-loading shotgun (such as the Remington LT 20) within your $300 budget. A 20 gauge Mossberg 500 (pump action) might be the best thing you can afford. If the stock's too long, having it shortened is a fairly cheap and straight forward modification.
BTW, Mossberg makes a .410 "Home Defense" shotgun which is marketed to women. I would avoid it.