State-feedback controllers are frequently employed in Airborne Wind Energy systems to follow reference flight paths. In this paper we compute the region of attraction (ROA) of path-stabilizing feedback gains around a periodic trajectory as typically flown by a power generating kite. The feedback gains are obtained from applying a time-varying periodic Linear Quadratic Regulator to the system expressed in transversal coordinates. To compute the ROA we formulate Lyapunov stability conditions which we verify by Sum-of-Squares programs. These programs are posed as an optimization problem for which we compare various definitions of the cost function representing the size of the ROA. In a numeric case study we demonstrate that the maximization of an expanding ellipse inside of quartic Lyapunov functions leads to significantly larger verified ROAs compared to the standard approach of optimizing over the scaling of the sublevel set of a quadratic Lyapunov function.