I don't think there is any way to balance that field!
I agree, the pilot played the odds and won. Truth be known, most of us have done that sort of thing, but perhaps not at that extreme. Aspen in the summer in a Jet commander was pushing it as it was. With just enough fuel to get to an airport about a hundred miles away. But as with all of the twin jet engine aircraft, balanced field length was not the problem, it was second segment climb limitations that was the most difficult to achieve to stay legal.
In anything that requires most of the runway to get airbourne out of Courchevel, the trick is NOT to rotate while still on the downward slope, even if you have the speed to do it. Rotating just prior to where the slope levels off is a sure way to guarantee a tail strike. The guy in the clip did it perfectly, and I'm pretty sure he knew the place well.
I'll certainly take your word for this, as you seem well acquainted with that airport.
I've never been there, Aspen was bad enough as far as I am concerned.