There is a reason Hamiltonian mixing is default. Most other DFT codes would use density matrix mixing, but that's for periodic or finite structures. For open systems, it was found a long time ago that Hamiltonian mixing, contrary to popular belief at the time, works better.
So, yes, I do think that mixing on the Hamiltonian will have a beneficial influence on the convergence since all that's left is to get dH down, the system is physically converged.
And, in case you are already doing that, perhaps increasing the mixing parameter a bit might be a good idea. And, as mentioned above, since the total energy is converged, all is good, so perhaps you don't need such a strict convergence criterion as the default.