The first thing you need to do is to verify that your TV is charging the trailer battery. Hook up and turn off the TV ignition. Measure the TT battery voltage at the battery with a digital meter that shows two digits to the right of the decimal point (ie 12.05V). Harbor Freight frequently gives these away with any purchase if you have a coupon for the meter.
Start the TV engine and measure the battery voltage at the TT again. It should show over 13V, possibly up to 14.1V. If there's no change from when the TV was shut off, then you're not charaging the TT battery from the TV. Follow the previous posts' recommendations to figure out why.
Your residential fridge will probably draw 25A or more from the batteries when it's cooling, usually around 40% - 60% of the time depending on how often you open the door and how hot it is. 200A of 12V batteries will give you 4 hours of cooling before you draw a fully charged pair of 100AH batteries down to the 50% mark, the minimum charge you want to have left when you start recharging if you want maximum battery life.
Residential fridges are a great option but they do have their drawbacks for dry camping.