If the battery were the issue then it would not work on GAS either.
Most likely issue is the electric heat elewment this is located in the big metal tube above the gas burner It might have failed.
But that said I've heard that a old fashion electric lamp (not one of the LED types) I'd try 40 to 100 watts.. Get a caged socket and a short Extension cord/splitter (See link) and plug it in and put in the exterior compartment.
Also bypassing the door switch so the inside light stays on (If your firdge has a LAT switch (Low ambient Temp) this is what it does. One or both may help
I'd try outside first.
The link is the type of cord I'm talking about but I'm not recommending either way the specific cord or the retailer.
Click here for an example
