A standard checkout test for a microwave oven is to simply fill a coffee cup with water - it should boil within 3 minutes.
The common electrical system with your (assumed RV ammonia ) refrigerator, radio and E3 error is the 12 volt DC electrical system. This is also somewhat related to your AC power system in that the converter/charger for battery recharge and 12 Volt DC current source when connected to AC shorepower is AC powered (it converts 120 VAC to 12 VDC). I am thinking (like you) that there is some deficiency in your 12 Volt DC sources/wiring causing low voltage at the appliances malfunctioning. I am disappointed that good "RV Service Professionals" have replaced the expensive refrigerator 3 times, and not found the source of the problems if it is related to DC/AC power systems. You may have to go looking for a Good, Experienced, Certified RV service professional (on your own dime) to resolve these problems.
You CAN operate your refrigerator to test it by simply disconnecting the 12 volt DC power on the back, wiring in an outside source of 12 volt DC power (battery charger?), and connecting the refrigerator Ac power plug on the back to an extension cord to a source of power OTHER than your RV. You need to then turn on the refrigerator for a few days, put a thermometer in a glass of water in the refrig., and take readings. It will normally take 1-2 days to reach full cold. You don't need propane for this electrical only test of the cooling unit and associated electronics.
BTW, you should also be able to run your radio on an external source of 12 VDC power if you can access the power connections.
I would check out your batteries, converter/charger, thoroughly inspect your 12 VDC wiring feeling for hot places (poor connections - 12 VDC doesn't have enough voltage to shock dry skin), shaking wire connections looking for loose wiring.
