I'm like you and really think IF will make a huge difference. Check out Dr K Bernstein's Diabetes Solution. He fasts and talks about how he manages it through injecting insulin. It's in chapter 6 under the heading the dawn phenomenon.
I may have similar issues but I need to mention that I'm low insulin and not injecting. I'm struggling with spikes after breaking the fast (due to even lower insulin), plus late night highs before bedtime (which must be gluconeogenesis, where the liver converts amino acids into glucose) which means I end up with high sugars all night and of course the dawn phenomenon where it takes all day to clear my morning high. I've found dirty fasting (cream in coffee) helps my break-fast spike, weights at the gym an hour before bedtime helps with the gluconeogenesis and dawn phenomenon. Fingers crossed as I've had 3 weeks of miserable blood sugars because I'm determined to persevere.