I do not know Harley's.
When you say it dies dead, but starts back up, I think electrical. If it was fuel, you would have a dry carb, and it would be hard restarting.
Do you have a side stand saftey switch? Could you be hitting a bump and the side stand bounces off the switch and kills the bike?
Or a short in the kill switch or side stand safety switch wiring. Look all along the wiring from both switches. Bikes are known to rub wires raw around the steering head and under the gas tank from the handle bars moving the wires back and forth all the time. Is all the wiring in the head light shell? If so, there is another rub spot.
I am really enjoying your trip. I wish you guys all the best.