Every time I have seen power stairs done, the athlete lifts the object to the next stair, puts it down there, and then goes up the stair. Repeat until you are at the top. Run down and get the next implement (if applicable). So don't think of it as going up the stairs while carrying something. Think of it as repeatedly lifting an implement (probably a duck walk implement) onto a small platform. Climbing the stair yourself is somewhat incidental (a little bit of cardio to go with the lifting).
With that in mind, straddle deadlifts to train. Heavy kettlebells if you have them. Farmer walk implements. If you have a duck walk implement, that is what is generally used. You'll have to consider the size of the implement and the height of the stairs to determine if you can cut the deadlift short or need to hyper-extend in order to get it onto the stair.
Falling probably shouldn't be an especial issue with this event. That said, people fall on much simpler events all the time, so there is certainly the possibility, especially with adrenaline, the clock, and pressure in a competition.