I can not speak about the needs of the 'Fridge.. But I can tell you that the list of things that does not work well with True Sine Wave inverters is kind of short.. In fact here is the entire list:
Anything needing more power than the Inverter can produce.
That's the whole list
Now with MSW you have a much longer list..
Anything using power line based timers
Audio, Video, Radio, and Television hardware
Electric Blankets, Night lights, Coffee pots (With electronic controls)
Some motors
Many electronic devices not listed above
Volt meters
And more.. yes it may work well.. Or you may not be aware that the problem you have is due to MSW inverter.. But Well.. I've had some serious training in that area.