Power supply for off-grid
The answer is that it depends on your load rather than the size of your generator.
Typically if you divide your generator full load gallons/hr by 3 you get your quarter power fuel consumption. Below quarter load on the generator the fuel wastage (just to keep the generator turning, and produce lots of unused heat) ratio increases for all engine driven generators.  

So if your cabin consumes 3000/4 = 750 W or less then you are typically wasting more fuel (the percentage varies with generator speed, rotating mass etc). Obviously as you approach no load (say 100 W or so), virtually all the fuel you consume is a waste. 

If you implement a battery/inverter solution and you run at say 750 W or less, then you will definitely save fuel. And if you run close to no load (say 100 W or so) on the invertor you will save significant fuel.
You should ensure that you get the largest possible charger for your battery pack, so that when the generator does run, it charges the batteries in the smallest time and highest load possible (most fuel efficient). 

You might consider also that the equation for generator design has changed in recent years. Some manufacturers now produce generators with DC-AC invertors inside them. These generators can slow down when lightly loaded and speed up as the load increases. This significantly reduces the waste fuel so might also be a potential solution for you.