Burning man is exactly right - the power from a modified square wave can be hard on anything connected that has magnetics (a transformer) in it. The result may be that the charge output from the converter may be lower powered from your inverter than from the wall plug. It also may not last as long. You would need to measure the charge current to see. An absorption refrigerator is a resistive load and would deal with it fine.
Less well known is that the output from RV type small generators is also ****. The waveform looks like a horribly distorted sine wave.