I'm confused.
You go high if you don't eat and walk?
And only a c-peptide test?
No other tests than the obvious blood panel for hba1c?
Have you an endocrinologist?
As per forum rules I cannot diagnose, however, by your description and my experience, this is not RH.
RH is caused by food. There are other conditions that have episodes of hypoglycaemia, they, if not T1 or Lada, are usually pancreatic conditions, such as insulinoma.
But I could be wrong..
I would advise you, to go back to your surgery, and ask your doctor for a referral to a specialist.
I will tag
@EllieM to pass on more about why you are having to use insulin.
Best wishes.