If you haven't tested your basal rate by fasting, that would be my starting point, as this can change over time, as can insulin to carb ratio. Weather, stress, illness etc all affect how much insulin we need, and when we need it.
Have your team suggested a pump at all, as that sorted my dawn phenomenon out. I know some people set an alarm clock and inject a small amount of bolus insulin during the night, but this obviously interrupts their sleep.
Other than that maybe a different basal/bolus insulin?