You probably have a MSW inverter. Although it may run the fridge, its not the most effecent type for electric motors. Also, some of the large inverters use nearly the same AH capacity just being on as the fridge will use.
My preference would be a dedicated 1000 watt, Pure Sine wave inverter to run the fridge. It won't draw more then 20 to 25 amps DC, when running the fridge. Your alternator will easily handle that. Keep the large, power hungry inverter off when off grid.
As far as wiring it, you could get one with a pass thru switch or use an external, inverter style transfer switch. They get wired into the fridge outlet wire and plug into the small inverter.
You could also just let the fridge run on the stand alone inverter full time. If on shore power, the battery charging will be handled by your original systems.