again i am not an expert. but from what you say all works just fine when all you have running off the inverter is the refrigerator. that fuse blows when you try to add the microwave. and of course when the fuse blows the converter panel goes out as it is not getting any power.
it just sounds like the microwave and refrigerator combo are overloading the converter and fuse. and power required by the refrigerator varies with time depending upon whether it has the compressor on, ice maker, defrost, etc. and you do not have any control over when these cycles occur. you might be able to turn the ice maker and defrost cycles off. and depending upon the functions active with the refrigerator at the time you add the microwave it may or may not overload the inverter. so it might work one time and not another.
then only thought i would have would be to go to the inverter and study its status display panel. most will show something like dc volts / amps in and ac volts / amps out.
take a look at what it shows with just the refrigerator on and have somebody looking at it as the microwave is turned on. you might catch some info on the display panel before that fuse blows. but the fuse blowing says you are drawing more power that the circuit is designed to carry.
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