The size of the wire is for the amount of current. If you are only doing a small amount of current, small wire is ok. You step up the voltage to get more power through the wire (power is volts times amps you step up the voltage the current goes down proportionally). You have 240 Watts of generator at 240 volts, 1 amp, or 120 volts, 2 amps.
So you run the amount of current through the voltage drop calculator until you are not below 10%. Or just post the KVA of the generator (or nameplate). We will figure it out for you.
The smaller the wire will just throw away more power in the form of heat.
The advantage of AC over DC is that you can step up/down the voltage with a transformer. It is not that it goes further.
The advantage of DC over AC is that there is no impedance. We step up the voltage to 500,000 volts and then convert it to DC to send it from Bonneville dams up here to California and from the CA nuclear plants back this way when they have a surplus.
Now I am going to get into the weeds. If the voltage drop is too large you can look at a buck-boost transformer. Basically it is a 120volt to 12 or 12 volt transformer that you wire in series.