This letter shows for the first time why it is important and how to optimize the gains of a position controller on board of a fully-actuated aerial vehicle with bounded lateral force, via an auto-tuning approach. In such vehicles, most of the control authority is expressed along a principal thrust direction, while along the lateral directions smaller forces can be exploited to achieve full-pose tracking. The nonlinear and hard to model interplay between the constraint imposed on the lateral force and the gains of the position controller is overcome employing the OPTIM-tune calibration method. Several experimental tests, performed fully autonomously during flight, clearly show the practicability and benefits of the approach.