As to the meter, I actually found the manual online and the only relevant thing was "Corrected mean corresponds to the sinusoidal RMS".

Not having a scope or the knowledge how to use one, I tried another test. I plugged in a 230vac to 24vac transformer in parallel with a 60w tungsten bulb and compared the no-load voltage of the step down transformer. It was practically the same, as was the observable bulb brightness, between running off the mains or the inverter. So, I guess I'm OK. The fan test is still worrisome for me though.