dometic refrig while traveling

As for fueling, for situations where safety requires that the propane be turned off, there is no point in running on battery for those few minutes (and it also has spark potential in the switches).  The fridge is well enough insulated that it will hold it's cold temp for way more time that it takes to refuel.  And if the flame from the fridge is a fire danger, so is the ignition switch and other electric components in the TV that can also provide an ignition spark for fuel vapor.  And they're a lot closer to the vapor source than the fridge flame.  i don't think any common TV's are manufactured to mine safety spark specifications, but I've been wrong before.

Sparks that can ignite vapor while fueling can come from pretty much anywhere, including the TV, a cell phone, the switch supplying energy to the electric heating element of the fridge, or just good old static electricity.  Given the very few explosions around gas pumps, these events are statistically insignificant [I found no reports of such explosions on Google, and if it ain't in Google it doesn't exist].  The same is true with explosions around gas pumps due to a  travel trailer fridge running on propane.  They just don't happen often enough to be an issue of real concern.  If someone has spilt a bunch of gasoline in the pump area, you are in as great a danger of a fire just driving in as you are with your fridge running as you pull to the pump.  Moral of the story, if you see spilled gas, don't drive near it, with or without a trailer.

The question is not one of safety but what kind of hydrocarbon you want to burn to power your refrigerator.  No one has discussed how much energy it takes to generate the electricity from your TV and send it down copper wires to heat a heating element in the refrigerator.  Do you use more energy burning the propane than gasoline to turn the alternator of your TV?  

My intuition tells me that burning gas to turn the flywheel of an internal combustion engine, which in turn drives a pulley belt, which in turn has to spin the bearings in the alternator, then sending those excited electrons down the copper wire with what ever ohms of resistance, depending on wire size, is likely to be less efficient.  But if we really want to fully understand the energy consumption, we have to go back to when the petroleum upon which the propane and gasoline were based.  What energy was expended to pump and refine them?  Which used less energy getting to the point of distribution.  Does the gas have ethanol, which has a negative energy balance [that is it takes more energy to produce it that it yields]?  How much extra energy did you consume going to Tractor Supply to buy the propane?  How many calories did it take to pressurize the propane in your tank and pump it from the mother tank?  

The point is there is no right or wrong answer because we don't know all the variables that go into producing a watt of energy for the fridge.  Advanced  planning to use electricity is not an issue of being prepared and using the most efficient system, it's a matter of being well prepared to send the electrons scurrying from your alternator to the fridge without burning up the wiring on your vehicles.  Using propane isn't any better because it, too, can present unplanned ignition and combustion issues.

For me, I don't want to spend the money on a heavy duty electric system for my little old Dodge Dakota with a stock alternator.  There is no discernible benefit in the cost/benefit analysis.  On the other hand, if I had a $65K Dodge diesel that came with an alternator big enough to light up Las Vegas, and 6 awg wire to a convenient plug in the back of the truck and my pod was wired with similar wire from the battery to plug into the truck wire, I'd probably use it, notwithstanding the ultimate energy use calculation.  [edit. - I forgot to mention my dream truck would have a 15 amp AC plug, so I could run the refer on AC and not worry about 6 awg. wire.]

So to everyone, use what you have, enjoy it and don't look down on the other guy who likes to do something different.  At the end of the day [an over used cliché], we all want a cold refrigerator so the beer doesn't get warm.