My endo is having my c-peptide tested (I just have to go do it one morning) but he's having it tested 30-45mins post breakfast to see what the insulin response is like to food. Not to see what the fasting is because he said it doesn't tell him much of anything in my case because my pancreas doesn't produce enough insulin. If you are low carbing, then the fasting test isn't going to show much if there is an insulin production problem, because it will be at it's lowest point in the morning and that's when demand is also at it's lowest point anyway. Plus low carbing creates the least amount of demand for insulin, but if your pancreas isn't making enough because of some kind of damage then what's the point of testing it fasted? IMO - you want to see how well it produces when it's in full swing, post meal.
As also mentioned, get the full antibody panel tests done if you can. I'm pretty sure you'll get an answer. I live with a pancreas that doesn't produce enough, which requires small amounts of insulin to make up for the shortfall. Not a big deal, but it's best to get the right diagnosis decided once and for all so you get the optimal treatment for it.
Just my thoughts from someone who has a part time pancreas. Good luck.