I would share your concern that your child could be developing type 1 diabetes. Children's blood sugars are typically lower than adults, and a result in the high 50s or 60s can be 'normal' for a fasting level. People without diabetes (or risk for diabetes) will typically never go above 120 (and usually their blood sugar would be tightly controlled in a narrow range all the time). The highs that you are seeing are concerning, particularly that he didn't have much carbs.
Putting him on a high protein, lower carb diet, with adequate healthy fats for energy may help avoid the highs and also help keep his sugars stable. Given that his lows are not 'very low" for a child, you may also try using less glucose for treatment (ie. just 1 - 2 grams, not 15 grams), and then follow this with a small protein snack.
Please request his doctor to do antibody testing. They should also testing c-peptide, and insulin levels.
Given his family history, you could also look up "trialnet", and then contact them. They will conduct testing of children of relatives with type 1. It is part of a large population study. They test for about 5 different antibodies and there are prevention / delay diabetes trials also being run.
I would suspect that he has early type 1 diabetes, and you are watching this develop 'slowly' over time.
Please check out Bernstein Diabetes University on U-tube. They have a lot of information and also "what to do" if your child is developed with type 1. They do use low carb, high protein diet as part of management.
Please let me know how you go. You may also message me, if you need further support.