Yeah, that works, if you kill the alarm and if the unit you have will switch to battery mode without first having ac power. In fact, it's what I used for about a week before the inverter I purchased arrived when I first put my pc in the car.
It's just easier and really quite cheap to have a modified sine wave inverter. And about the 60HZ wave being cleaner? It doesn't really matter. Power supplies now are so advanced that they don't care about a jagged, messy input.