Not as a general principle. That doesn't mean that constructivists say that there can't be sets for which the operation is valid. In particular enumeration is not a precondition for a thing to be pickable.

Now they say that the levers are indistinguishable, which means that their difference actually does not lie in their identities, but their relationship to the space they're in (or everything would collapse into itself), thus I don't have to look at the levers I can look at the space. They say that "I can't enumerate them all" but that means that there's at least a countably infinite number of them.

So the solution is easy: I take the space, throw away all of it that doesn't hold a that countably infinite subset, observe that the result is now isomorphic to the naturals, then cut it down to six, and throw a dice. There, not just arbitrary but even (a bit) random.

Really, only ultrafinitists would have trouble with this... but then they'd turn it around and challenge you to actually construct that infinite number of levers for real, not just in the abstract, and untie everyone while you've stopped the tram due to being caught in an endless loop.