If it were me, I would alternate. The benefit of a stair climber is its all up all the way, but the downside is, presumably, you don’t plan to exclusively ascend to basecamp, I assume you’d like to return. Also, the stairclimber provides a mechanical advantage in that it lowers the stairs to you. The benefit of your house stairs is that you actually have to lift your leg the whole way to the next step with no mechanical advantage and you have to train going down, but the downside is you’re going to be going down really frequently which does give your muscles a break.
So my inclination is to do them both.