Let me preface this by saying that I'm not a refrigerator restorer (although I do have a post war GE single door that works fine and I plan on repainting and putting back into service someday). What I'm about to say is based on what I discovered researching your question and my own experience with various chemicals.
Both sulfur dioxide and methyl formate are toxic. Methyl formate has the additional feature of being extremely flammable, to the point of being explosive. It's also narcotic, which I guess might be a benefit if it knocks you out before it explodes and burns you to a crisp.
In any case, you might want to contact one of the antique refrigerator restoration outfits (there's one in Georgia) and ask them what they'd do. In general I gather that if you were to be able to find someone to recharge the thing, you'd first want to make sure the source of the leak is fixed, cause obviously you don't want a toxic and explosive chemical wafting through your home. I gather that a main cause of leaks in early GE monitor tops was corrosion of a steel part (compressor shaft?), which GE later corrected by using stainless steel instead. Then again, sulfur dioxide is also rather toxic but there seem to be a fair number of old monitor tops still in operation that presumably still have the original refrigerant (SO2 or Methyl Formate).
I also gather that in some cases the restorer will upgrade the entire refrigeration system to something more modern, using, say, R134a instead of the older toxic hazardous stuff.
If it were my fridge, I'd probably contact a licensed pro to check out the system and recharge it, if it's even possible. Methyl formate seems to be a fairly common precursor to various other chemicals, so it's still being made, just probably not expressly for refrigeration purposes.