Your Salem's residential fridge, once cold, will run intermittently to keep itself cold. Assuming your house batteries are fully charged and you have an active charge line from your tow vehicle, this should work without problems. Unfortunately, the same isn't true if the fridge isn't cold to start with.
When the fridge is warm, the cooling unit runs continuously, with a fairly large draw on the batteries. The charge circuit from your TV can't supply enough continuous current to match the drain rate so you'll be slowly discharging the batteries until the fridge comes to operating temperature and starts cycling. Depending on the length of your drive, you could end up arriving at your destination with nearly discharged batteries.
I'd strongly recommend measuring the current draw on the batteries when the fridge is drawing power to see how long the batteries will run the fridge. Also check the charge line from the TV to see how much current it will supply to the rig when you're towing it. You may want to install an additional larger wire charge line from the alternator to the rig, possibly bypassing the standard connector between them so you can effectively power the inverter from the TV's alternator.
Residential fridges are becoming more common in all types of RVs. I believe that the much lower cost than propane and the more "home like" cooling capacity are major factors driving this change. Don't forget that most current residential fridges, when full of food and cold, should be able to go at least 6 hours without warming up more than a few degrees if the doors are kept closed.
Phil
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