While the gen is putting a 12DC charge on the battery whatever DC charge gauge you are reading will read full, this is because the generator is putting full voltage on the battery and the gauge sees this as a full battery, remove the generator and your battery will go back to only a partially charged voltage and this will show on the gauge, but sometime there is a "surface charge" that will sit on the battery, some kind of electrical mystery, but that surface charge will go away after just a very short time of using lights, within minutes.
The generator probably only provdes very minimal DC power (amps), so will take a very very long time/ multiple tanks of gas to recharge two low batteries.
Better to plug a good battery charger into the AC plug of the generator and charge the batteries off the charger.
Solar is really the way to go. You batteries will last a year or two longer, maybe 3 or 4 years longer since you are hard on them running them down like you described. 100 watts will suffice and 200 watts is plenty. They key here is the charge controller (takes current from the solar panels and controls what goes to the battery). MPPT is what you want not the $20 ebay one, but at least $50 range will do. 20-30 amp model is enough for what you describe. You batteries will be topped off nearly all the time.