For someone who is a diabetic a reading of 87 mg/dl (4.7 mmol/l) is rather low. I am not surprised that you are having the symptoms of hypoglycaemia.
Blood glucose typically varies from 4 mmol/L (72 mg/dl) to 6 mmol/L (108 mg/dl) for people without diabetes.
Be guided by your feelings and if you experience these symptoms which indicate a low sugar level, then have something to eat. You do not need to have sugar, just a complex carbohydrate, even a piece of apple, brown bread, some salad like carrot.
You know how you feel, and if you get the shakes and feel weak, your body is giving you this signal that you sugar levels are too low and it requires some energy.
Sugar levels that drop too low can lead to a coma.
If you can bear with it stick with the diet. This is bringing your sugars back to where they should be. You will be much healthier if you can stick with this long term.
Your feelings of being shakey will pass, and important that you know your reading is now normal.
You are having a false low your body is used to high numbers so a good healthy number feels low. as your body gets used to normal BG the false lows will go away.
you can do whatever you want, its your body, you will pay the price for high BG.