As I understand it, your own, natural insulin is secreted in tiny bursts, carefully calculated to keep your blood sugar at a perfect level. Somehow I doubt if your body, on getting an injection of insulin, will say, "oh goody, I can doze off for a while". It's more likely that you will simply have too much circulating insulin. This will cause weight gain, low blood sugars and insulin resistance as the body's mechanism for protecting itself.
I am always in awe of the T1 diabetics who manage to maintain near normal blood sugar levels. They are constantly having to balance and calculate, taking into account exercise, activity, temperature, illnesses, stress, never mind what they have eaten. To put yourself in a position of having to calculate and compensate for injected insulin and take on board the legal requirements (driving), would seem to me to be madness, if you can avoid it. If you have beta cells that work, but feel you should give them a rest, simply cut right down on your carb intake.
Sally