Can you ride the bike on its lowest power setting?
I haven't been riding an E-Bike, but it sounds a lot like battery headlights that blink off
What I would do is get a simple multimeter.
Wire it directly to the main +/- terminals of your battery and add enough wire to get it to the handlebars.
Then ride your bike and watch the voltage (DC).
I'd anticipate that you'll see significant battery sag before it cuts off.