The frig uses very little energy, but the LP system is about 80% efficient. The genny, at best, is about 20-25% efficient.
So the direct answer is that LP will use less fuel, but if you need the genny for other purposes, the frig load on the genny won't be much. Take your choice.
Another consideration might be the availability of fuel for each. It's easy to get fuel for a non LP genny, but finding propane could be a problem.
As a general rule, using propane at 80% efficiency for the same job always uses less fuel than a genny that must convert fuel into electrical energy with an efficiency in the 20 percent range and then convert that into heat.
2007 Newmar Kountry Star DP
Cummins ISL, Cummins E-Brake