As others have said, don't even think about drawing 27AHr (300WHr) from the 60AHr AGM battery. And since it's a compressor, there will be a large turn on surge.
Now if you could assure that solar panels would provide a large part of the 300WHr, could be an interesting project. I've done similar. But if you have a 30W panel in the windshield (inside the car), you won't get anything near that most of the time. Best case, you may get 20W for a few hours (60-100WHrs?). If you have panels in both front and back, only one will be getting significant sun. And the back glass is 70% transmission from the factory.
If you go with a separate deep cycle battery, the solar may still make sense to help maintain the battery. Don't connect a panel of more than 10-20W directly to a 12V battery, as most panels can produce up to 20V+ in full sun. Standard solar charge controllers have a separate load output, which you connect the fridge to, and also protects the battery from discharging too low. But you may need a 30A controller to handle the compressor surges. If you're not worried about discharging the deep cycle battery (because you'll be back in 10 hrs max), the fridge can be directly connected to the battery, allowing you to use a smaller (10A) charge controller.