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Poincaré maps are a basic dynamical systems tool yielding information about the geometric structure of the phase space of the system. Poincaré maps are however time consuming to compute. In this paper we have analysed and compared two different schemes to compute Poincaré maps in the context of accuracy versus computation time: a Runge-Kutta method of 7th and 8th order and a time transformed geometric method of 6th order. The dynamical system used is the Restricted Full Three Body Problem, with the primaries, an elongated body and a sphere, in a short axis relative equilibrium configuration. Using these Poincar´e maps we have studied the dynamics near the collinear Lagrange point L3, located on the outer side of the elongated body. We present evidence that the L3 point in this system can have saddle-center, stable or complex unstable behaviour depending on system parameters. We further show that when a low accuracy regime that still captures the correct structure of the Poincaré map is considered, the geometric method clearly outperforms the Runge-Kutta method being up to 4 times faster to compute and free from accumulating local errors that smear the structure of the Poincaré maps.