Actually, a propane fridge will run 'off grid' for a long time. It still requires a 12v source to run the circuit board, but pulls very very little power since it is using propane for a heat source, in relation to a 12v compressor fridge - my trailer's propane I think pulls only 2a on gas, but the 12v pulls something like 7.5a. Big difference.... A propane fridge with 2 6v's can easily last a week. The problem is that the standard RV industry propane fridges also have major flaws, and have a difficult time keeping internal temps down to what a fridge should be. At least thats the case I have experienced, and my neighbor happens to be a RV tech, and said that's pretty much the norm, as wrong as that sounds... So it's a catch 22 really. A compromise - a propane fridge will last longer than a 12v compressor fridge 'off grid'; however, a 12v compressor fridge will actually get to appropriate temps....
Now some issues I find with battery power, is in theory you are correct about the amount of 'power' storage you need. But that's assuming that after 3 days you have the abilities to fully charge those batteries and/or those solar panels are putting out their max potential. The reality is 1) only having a hookup will actually do that - or running a genset for an entire freaking day (lots of gas), 2) I find it's actually a rarity that solar panels put out as much as desired even though they are still an excellent way to go, and 3) its just not practical unless your only doing 2-3 day trips, in my opinion. I am finding out, that the guys who are using monster trucks (think MAN) for world expeditions have massive battery banks, but still have to charge them via shore power somehow some way, or a genset, every several days depending on how long they stay somewhere. So I think though theory works, the realities are that its not really very practical unless you have a significantly larger battery bank. The other thing to consider that I find is what other items will you have running? Even something as simple as a couple lights, or a 12v water pump can put your 12v fridge below the auto-shutoff threshold. And thats not mentioning how much power you need to reserve for starting the truck. In this instance, though not everyone does it, I personally would find it a necessity to have batteries running the fridge or any other 'house' item to be isolated from the main vehicle batteries, except for when driving/charging.
Now I'm no expert, but these are things I have found out and experienced that have turned me away from a vehicle mounted 'house' (ie a camper) and steered me more toward a reasonably sized (21-26') trailer. That way you can still, say, have a small 12v fridge in the truck that only runs to keep lunch and beverages cool, but the trailer can have the capacity for a genset and a very large battery bank since that would be some pretty significant weight. But even then, the more I look into the real world power requirements of a 12v fridge, and considering I'm full-time, I honestly must say that a propane fridge, even though fragile, is looking to be pretty darn enticing, if only to keep the weight of batteries down. The last thing I'll mention now is the consideration of cost. Sure propane costs a fair amount. But 4 30# tanks can last a good while depending on other uses. But to get a battery bank that could last a significant amount of 'off grid' time in the dead of winter (I live in Montana), your talking about some massive battery banks, and those suckers are super heavy and expensive!
If only someone made a diesel fridge!!!!! I think in the end it may depend heavily on your own personal use or expected use. If you just want a couple day camping trip, then 12v is probably just fine. But if you live full time in your rig, and want to keep things running to keep food good when away for weeks at a time due to work (like me), or to even simply just boondock for a good long while, then battery power without some sort of shore charging or a genset every few days is going to be pretty darn difficult.
But back to the OP - since your staying in your trailer, and primarily using your propane fridge, and really just using the ARB for day trips to keep lunch cool, you might just need one extra battery in the truck to support the fridge for the few hours your out hiking or doing what-ever, so that you dont draw down the truck batteries. Then you could charge that when returning to the trailer with either shore or a genset to top it off, since driving likely wont charge it fully in itself.
