In the last years, the Pyramid WFS finally proved itself to be a very powerful tool for wavefront retrieval, in different applications, inside and outside Astronomy, often showing outstanding results. However, being intrinsically a non-linear WFS, the P-WFS non-linearity error starts to play a role when the AO loop is not closed on the sensor zero-WFE point. This led to the need to elaborate new concepts when trying to apply the P-WFS to open (or partially open) loop based techniques, not to trade sensitivity for linearity. This was the case for GMCAO, in which the reference stars are selected on a wide technical area of the sky, outside the FoV to be optimized, limiting the correction experienced by the WFSs to poor Strehl Ratio regime. While, in the recent past, we proposed a solution based on the Very Linear WFS, a sub-system that locally closes the loop on the Pyramid pin to let the sensor operate in its best regime, we now explore a different approach in which the P-WFS non-linearity is continuously measured, injecting a known aberration onto the sensor. In particular, we evaluate in this paper the possibility to apply basic PWFSs to the GMCAO technique, measuring the nonlinearity of the sensor and taking it into account in the wavefront computation, with an approach similar to what already proposed in the LBT AO facility FLAO for the non-common path aberrations correction.