Since you already HAVE that MSW inverter and evidently have been running that residential fridge on it which surprises me, as I'd expect such a unit to have a higher starting draw than a mere 1,000 watt inverter could deliver, but evidently I'm wrong about that in this case, either because your fridge doesn't draw as much as I'd expect or because your inverter has a higher surge capacity than I'd expect.
In any case, if you've been running it that way, and nothing bad has happened, I wouldn't worry about it. There are two parts to any fridge, the compressor or heating element in the case of an absorption RV fridge, and whatever controls said fridge has associated with the thermostat. Both the motor and the heating element should work without issue on MSW power as well as PSW. It is the controls that might have an issue, and only some of those would have trouble. Either way, trouble, if it were to occur, ought to happen fairly quickly, that is, within minutes, or at most a few hours, of beginning operation. If the fridge has a fairly simple thermostat to turn the compressor on and off, this should be no issue. It is pretty much only if the fridge is a fancy, multi-function unit that it ought to be an issue.