That's very Sherlockian, BJM! I'd never considered that, but for people who let their cats out, this is definitely a consideration.
I feed my cats a raw diet using the premix from Feline Future, which I've done for 14 years, and the owner of FF, who is wonderful, assured me that the diet I feed by using her premix "is as low in carbohydrates as you can go, containing 1% or less carbohydrates. Even a mouse has more! (2%). Traces of glycogen are found in liver and muscle meats, which is stored glucose for use as energy by the cells. It is sometimes referred to as "animal starch". There is no way to omit that from the diet, because all animals store glycogen in their liver and muscle tissue.
If you intend to increase the amount of fibre in Sam's diet, make sure you are not adding carbohydrates with that. Fibre is typically from plant source, and if you add vegetables, bran, flax seeds, psyllium, or similar, you will also add starches and other plant sugars to the diet.
Although cats eat a diet nearly void of carbohydrates, their cells need glucose for energy nonetheless. The main source of energy for cats comes from protein, and glucose is produced via gluconeogenesis during the breaking down of protein. It is a dysfunction of the adrenal gland or the pancreas which throws off Sam's blood glucose levels and not the carbohydrate content in his diet.
I don't think you can do anything dietary, except for, perhaps, trying supplements which may help regulate blood sugar or help restore adrenal and/or pancreas function. You will need the help of your Veterinarian to try and regulate Sam's blood sugar and determine if he has become insulin resistant or if his immune system is destroying insulin-producing cells in his pancreas."
I'm posting this here in case any of the other members find this information useful. The owner of Feline Future is a very caring person, and is a personal hero of mine for her work on biologically-appropriate diets for cats.
Anyway, I've got my Contour meter (thanks to Wendy&Tiggy) and the rest of the kit, and will go learn how to take care of Sam at the vet's later today.
Thank you everyone for your kindness, friendliness and support.