I think I'm right in saying, if the liver has pumped out extra glycogen during the day, whether to provide extra energy during exercise, or to bring Bg levels up after a hypo, it needs to replenish itself afterwards. it probably does this at night. I know I'm prone to hypos in the night if I've had extra exertions during the day.
There was a TV programme on weight loss a couple of years ago, that measured calorie usage, and it found that extra calories were still being burned 12 hours after exercise, and it attributed this to the body taking Glucose out of the blood stream during the night to replenish stores in the liver. in non diabetics, this wouldn't result in night time hypos, because insulin production would be reduced, but for those of us on long acting, the insulin is still churning out.
I know you're not on long acting. but could your general insulin response be impaired, so your pancreas doesn't get the correct signals to shut off production? ( I think this is where the c peptides and the other thing whose name escapes me that diabetics don't produce, may come into this process somewhere.)