Since the blood tests seem to confirm that you are producing lots of your own insulin, then the suggestion to up the injections is actually counter productive. It will flood the body with insulin and force more cells to store glucose, thus making your condition worse in the long run. As a fellow T2D who had similar high blood sugars 4 years ago, I was not on insulin but was about to be put on it when I tried Low Carb diet. For me there was an almost instant reduction in blood sugars and weight, and I was able to drop most of my medications. I am still not on insulin, I still low carb but no longer use keto diets I still produce my own insulin and it now works almost as nature intended, I am not cured, but I am in control.
This Christmas I stepped off the diet wagon and had a very enjoyable unrestricted christmas fare. My daily average bgl readings did rise a little from 6.9 to 7.7 mmol/l. Now 4 years ago that same fare would have put me up into the 20's and 30's, but the highest this year was 10.1 at the 2hr reading after the main meal with Xmas pud, mince pies and brandy butter. So as you have the insulin yourself, then a serious look at some of the Low Carb diets here in the Forum are worth considering again. Stop feeding the fire, then you need less to put it out.
Personally I now use the Pioppi diet, which is an LC variant of the mediterranean diet. It is not keto, but is a lifestyle I can follow easily. There is alternatuvely the Paleo diet, the Atkins Diet, or the reduced calorie diets like Newcastle or the Blood Sugar diet. They all seem to be viable for T2D