Conventional OFDM, adopted in LTE-A systems, cannot provide the quality of service requirements sought in 5G systems because of extreme natural channel impairments caused by higher Doppler spreads and unexpected artificial impairments caused by multi-source transmission, to be brought by 5G, and by synchronization relaxation for closed-loop signaling overhead reduction in some 5G applications. These severe impairments induce a strong loss of orthogonality between subcarriers and OFDM symbols and, therefore, lead to a dramatic increase in ICI and ISI. To be well armed against these dramatic impairments, we, in the present paper, optimize the transmit/receive waveforms for FBMC systems, with hexagonal time-frequency lattices, operating over severe doubly dispersive channels, accounting for both natural and artificial impairments. For this, we exploit the POPS paradigm, recently proposed for rectangular time-frequency lattices, to design offline waveforms maximizing the SINR for hexagonal time-frequency lattices. We show that FBMC, with hexagonal lattices, offers a strong improvement in SIR with respect to conventional OFDM and an improvement of 1dB with respect to POPS-FBMC, with classical rectangular lattices. Furthermore, we show that the hexagonal POPS-FBMC brings more robustness to frequency synchronization errors and offers a 10dB reduction in OOB emissions with respect to rectangular POPS-FBMC.