If you get a larger output inverter, you could concievably run the fridge for a brief burst (while drawing down the battery), and then shut off the inverter while the battery charges. The insulation in the fridge would keep food cold until the battery could get caught back up enough to run the inverter for another burst to keep the temp down.
Either way, you're going to want a bigger inverter for what you're talking about. Harbor freight has some good deals on them... I got a 3000w inverter for less than $150 a couple years ago. Might also be worth investing in a "dual battery" setup for your vehicle - it wouldn't increase your duty cycle, but it WOULD increase the time period of both the inverter draw and recharge time. I.E., instead of running the inverter for 10 mins, then recharging for 10 mins (switching back and forth every 10 mins), with two batteries you would only have to switch from draw to charge every 20 minutes. Would make it so you could do other stuff while doing this... instead of having to have someone "babysit" the setup and switch over constantly.