I have used the Progressive app on my phone as proof of insurance with officers. This IS actually standard practice in many states (Tennessee, for example, completely allows digital proof of insurance). Every time I have showed proof of insurance to an officer this way, they have never taken my phone back to their car. They saw that I had insurance and that was good enough for them and they didn't ask any other questions.
As for the "battery is dead" issue, that should be a non-issue in the car if you have a car charger for your phone.