Rational functions are able to approximate functions containing branch point singularities with a root-exponential convergence rate. These appear for example in the solution of boundary value problems on domains containing corners or edges. Results from Newman in 1964 indicate that the poles of the optimal rational approximant are exponentially clustered near the branch point singularities. Trefethen and collaborators use this knowledge to linearize the approximation problem by fixing the poles in advance, giving rise to the Lightning approximation. The resulting approximation set is however highly ill-conditioned, which raises the question of stability. We show that augmenting the approximation set with polynomial terms greatly improves stability. This observation leads to a decoupling of the approximation problem into two regimes, related to the singular and the smooth behaviour of the function. In addition, adding polynomial terms to the approximation set can result in a significant increase in convergence speed. The convergence rate is however very sensitive to the speed at which the clustered poles approach the singularity.