A Small dorm fridge will use 50 to 100% more electricity than a 12v compressor fridge of the same size, and require an inverter likely in the 1250 watt range to handle the start up surge of the compressor.
Depending on batteries, to power 12v appliances for regular daily cooking, is not wise. Heating anything uses a lot of electricity, and while batteries might be big and heavy and expensive, they just do not store much electricity.
Even with 200 watts of solar, your batteries are going to live in the very low state of charge, and far too soon they will not be able to power your fridge. The batteries will lose capacity quickly, and even after you find somewhere to plug in to recharge, they will exhibit this lost capacity from their abuse of being discharged and kept discharged while trying to power your 12v cooking appliances as well as your inefficient dorm fridge.
People with large battery banks and lots of solar and generators to recharge and to power larger loads, can get away with dorm fridges, or even full size residential fridges.
We all understand that dorm fridges can be picked up cheaply. Most who do so, do not realize the need for a larger inverter to power it, more battery to power the inverter, and more solar and other charging sources to replace the extra juice required to power the inefficient dorm fridge.
All that money initially saved by going with a cheap dorm style residential fridge, quickly gets eaten up by the bigger battery bank, the extra solar required, and other means to recharge( generator and plug in charger) that will be required to power just the fridge by battery alone. Once you Fall well short, add in the price of another set of batteries that you killed by chronic undercharging and overdischarging.
Once you throw in all the other things you will ask from your batteries, they are simply doomed, and you are doomed to frustration, and you will eventually come to realize the 120 dollars you spent on a dorm fridge was better spent toward a 12v compressor fridge, and that energy contained in one small green bottle of propane is about 20 or 30 AGM batteries worth. Though I have not done the math on that number, and won't, so don't quote me on that figure.
And yes we all understand the fear of cooking via burning hydrocarbons in an enclosed area. We are all aware of the risks. We take precautions and are still alive.
Cook with propane or butane, allow for forced air ventilation when doing so, get a 12v compressor fridge, and then 200AH of AGM and 200+ watts of solar will allow your batteries to live a respectable lifespan, and for you to live in this manner where you do not have to stress out about "do I have enough battery to cook with, and power my fridge? What if I don't and 60$ worth of groceries go bad? What if I get sick from eating spoiled food?
Hearing a low voltage alarm on the inverter at 3 in the morning, will make you wish you did it right the first time, and 'doing it right' and 'dorm fridge' are never spoken in the same sentence, Unless you will always be able to plug into the grid overnight, every night.