"what could cause reaccurent severe hypoglycemia in a type 2 diabetic who does not take any medicine or insulin and is diet controlled?"
This has been reported with teens as they grow rapidly but your photo indicates otherwise. You left out some important information to help make a determination as to the cause, such as when/time you're testing - prandial & postprandial - foods you are consuming, body weight, type of exercise and any medications you're taking not related to diabetes, including Aspirin, supplements and such.
Furthermore, have you tried going on a nutritional diet to see if the foods you are eat are causing this problem? e.g.; Eat the same foods, same proportions, at the same time for three days and test prandial & postprandial.
What is your A1c level? How is your Thyroid (TSH)?
What's interesting is you said you checked your "five" meters against your Endo's and she still doesn't believe you. My meter keeps a 30 day or 30 tests history [some only 10 days] stamped with time and dates. Did you show her this?
Unfortunately, it sounds like youd diabetes is not under control. I don't understand why your doctor doesn't believe this. If you get highs and lows despite your attempt to control your diet and exercise, it sounds as though you need further treatment such as oral medications. The lows are just a part of the cycle of uncontrolled diabetes: you go high and then you crash. I hadn't eaten sugar for 14 years when I was diagnosed with out of control diabetes! But your good diet habits are still very important because the meds don't do it all!