I would say propane most likely beats Grenerator but I"m not sure. Depends on the cost of fuels.
I will give you HALF the information you need.
Thomas Edison achieved, as I recall an efficiency of about 80% both on Generators and on electric motors. Though there may have been SOME improvement since. not a whole lot.
SO you burn Fuel (Gas D or P) it makes mechanical energy (There is some loss) Which then makes electricity (Again 20% loss) this heats a wire (At last we have something that APPROACHES 100% efficiency (There is a small loss in the wires leading to the heat element) to run the Absorption Cooling Boiler
Now think of Mr. Rube Goldberg.
Next we have the propane method...
You burn propane to make heat which heats the boiler.. End of list.
That said. If you are paying like 5.00/gallon for propane and 2.50 for gas.. Well makes a difference don't you know .