You have reversed the 2 factors you wrote, due to which fuel efficiency is affected.
Fan speed doesnt require power from engine directly, but the generator\battery. So it doesn't alone affect the fuel efficiency.
The compressor does affect fuel efficiency directly.
It indeed works at full power always, and only it's On timings are different wrt the temperature thats been set. By lowering temperature setting i hope you mean, increase the cooling i.e. 16C (in terms of our home ACs). So by lowering the temps you are actually asking the compressor to work more to achieve that temperature, meaning the compressor stays on for longer time, hence consuming more fuel. Pls know that compressor is directly connected to the engine and takes power directly from engine.
Now, note that the higher the fan spd, the longer the compressor will run (assuming temperature is in coolest position) - hence more fuel consumption...... and vice versa.
You can make multiple settings to save on fuel.
I mostly set fan at 3spd but temperature at like 26C, as i like more air volume than cooling effect. So despite fan is higher, fuel efficiency wont get high cause i increased the temperature of AC (decreased the cooling)