You do not need a generator to run a small 12VDC refrigerator. I have a 2018 Monterey 335 SY that has 2 refrigerators (Isotherm 65 and 49, both ). I have let them both run, on the highest setting, in my driveway without shorepower plugged in. I have two group 31 AGM batteries for the house loads. After a full 48 hours, the batteries were still at 11.9-12.0 volts. This was about a month ago when it was 95 degrees out, which means it was easily 105+ in the cabin and covered cockpit. The refrigerators are rated for 109 degrees (Tropical version) and were maintaining about 37 degrees according to the wireless thermometers I keep in them.
With a small refrigerator and a good group 31 AGM battery, you will be perfectly fine for the day. The issue you will run into is that they do not cool down instantly (I never shut mine off, they are on AC power in my storage hanger and when I trailer to a lake, I leave them on DC power until I get launched and the generator started). So you will want to have a good battery charger so you can leave it plugged in when the boat is stored in order for the refrigerator to be ready to take cold food when you head out for the day.