I live off grid and in the winter and when the solar needs a boost i run a small generator with AVR ( not an inverter generator ) and a 3 stage battery charger like shown below, its sited away from my accommodation so I can't hear it but the tank is very small on it and runs dry well before charging is completed, I'd like to know if the surging and erratics rpm of the generator as it runs out if fuel will damage electronics like this charger or the AVR on the generator , it would save me a trip outside at night if I could just leave the charger on until the generator runs out of fuel and splutters to a stop.

 

Hard to say for sure.
The voltage will likely just decay along with the frequency.
The only problem I see is if the any of the connected devices do not like the low frequency.

If it's already run dry and stopped with no problem, then I would think it's okay to let it do that all the time.

If you wanted to get fancy, you could add a low-voltage cutoff relay at the output.

 

I would check the voltage when its surging, if it is a true AVR regulated it should handle it. If its actualy a Brushless generator then the voltage can go quite high when the engine RPM goes high. I have seen it go to 300V on a 240V generator.