Over the last decade, significant work has been carried out in order to increase the energy/peak power provided by fiber lasers. Indeed, new microstructured fibers with large (or very large) mode area cores (LMA) such as Distributed Mode Filtering (DMF) fibers and Large-Pitch Fibers (LPF) have been developed to address this concern. These technologies have allowed diffraction-limited emission with core diameters higher than 80 μm, and have state-of-the-art performances in terms of pulse energy or peak power while keeping an excellent spatial beam quality. Although these fibers were designed to reach high power levels while maintaining a single transverse mode propagation, power scaling becomes quickly limited by the onset of transverse modal instabilities (TMI). This effect suddenly arises when a certain average power threshold is exceeded, drastically degrading the emitted beam quality. In this work, we investigate the influence of the core dimensions and the refractive index mismatch between the active core and the background cladding material, on the TMI power threshold in rod-type Fully-Aperiodic-LPF. This fiber structure was specifically designed to enhance the higher-order modes (HOMs) delocalization out of the gain region and thus push further the onset of modal instabilities. Using a 400W pump diode at 976 nm, the power scaling, as well as the spatial beam quality and its temporal behavior were investigated in laser configuration, which theoretically provides a lower TMI power threshold than the amplifier one due to the lack of selective excitation of the fundamental mode.