Why does the result of the Lagrangian depend on the formulation of the constraint? Consider the following maximization problem:
$$ \max f(x) = 3 x^3 - 3 x^2, s.t. g(x) = (3-x)^3 \ge 0 $$
Now it's obvious that the maximum is obtained at $ x =3 $. In this point, however, the constraint qualification
$$ Dg(x) = -3 (3-x)^2 = 0$$
fails, so it's not a solution of the Lagrangian.
Re-formulating the constraint as
$$ h(x) = 3 - x \ge 0 $$
allows obtaining the result, as the constraint qualification holds: $ Dh(x) = -1 $
Now, I'm well aware that the Lagrangian method can fail under certain circumstances. However, isn't it kind of odd that re-formulation of the constraint yiels a solution? Does this mean that whenever we're stuck with a constraint qualificiation issue, we should try to 'fix' the constraint?

Yes, reformulating the constraint might change the validity of the CQ. This is quite natural, since the Lagrangian expresses optimality via derivatives of the constraints (and the objective). Changing the constraint may render its derivative useless ($0$ in your case).
