A 3-way fridge, also known as an absorption fridge, uses heat to provide energy needed to drive the cooling system of the fridge. It is hugely inefficient to use 12 volts from your truck or camper to provide that heat just like you'd never use your 12 volt system to run an electric heater to keep your camper warm inside. Here's why:
A gallon of propane can produce 91,502 BTUs, a measure of heat, or energy.
To compare propane to a battery, you have to calculate how many BTUs of heat a certain battery will produce. A KWh of electricity is equivalent to 3412 BTUs. So there's the equivalent of 27 KWh of electricity in a gallon of propane (91,502/3412).
A typical 12 volt camper battery is rated at 80 Amp-hours. Since watts roughly equals volts X amps, a camper battery (remember, you only want to discharge no more that 50%) can practically produce only about 0.5 KWh, something like the equivalent of 1700 BTUs.
Bottom line, there's 54 times the energy stored in a gallon of propane than can practically be obtained from a typical 80Ah camper battery. No wonder we use propane, rather than electricity, for heat (unless, of course, we're connected to shore power).