Dear Lorenzo,
Regarding the oscillations, you may need to decrease EPS_DEFAULT and EPS_SCHWARZ. With OT, you may also try other preconditioners and try to lower ENERGY_GAP in the OT section. I am also not sure whether PBE-pseudopotentials are suitable with HSE06 and should be replaced with pseudopotentials optimized for hybrid functionals (PBE0).
HSE06 is generally more expensive than PBE but you may try to set SCREEN_ON_INITIAL_P T in the HF section (at the expense of a lower EPS_SCHWARZ) to decrease the number of necessary integrals. The runtime of the second step should be significantly lower than the first SCF-step. To achieve that, you should set the MAX_MEMORY keyword and increase its value to fit as many integrals into RAM as possible. If there are still too many integrals to be recalculated, you can reduce the number of MPI ranks while increasing the number of OpenMP threads per MPI rank (and MAX_MEMORY).
It may also help to increase the print level and as soon as the accuracy looks fine, to post a timing report of a few SCF steps for further advice by either using a larger value of EPS_SCF or adding
IGNORE_CONVERGENCE_FAILURE to the SCF section.
Best,
Frederick